Triangle. Congruence

Transcription

1 Triangle Congruence 4A Triangles and Congruence 4-1 Classifying Triangles Lab Develop the Triangle Sum Theorem 4-2 Angle Relationships in Triangles 4-3 Congruent Triangles 4B Proving Triangle Congruence Lab Explore SSS and SAS Triangle Congruence 4-4 Triangle Congruence: SSS and SAS Lab Predict Other Triangle Congruence Relationships 4-5 Triangle Congruence: ASA, AAS, and HL 4-6 Triangle Congruence: CPCTC 4-7 Introduction to Coordinate Proof 4-8 Isosceles and Equilateral Triangles Ext Proving Constructions Valid KEYWORD: MG7 ChProj Congruent triangles can be seen in the structural design of the houses on Alamo Square. Alamo Square San Francisco, CA 212 Chapter 4

2 Vocabulary Match each term on the left with a definition on the right. 1. acute angle A. a statement that is accepted as true without proof 2. congruent segments B. an angle that measures greater than 90 and less than obtuse angle C. a statement that you can prove 4. postulate D. segments that have the same length 5. triangle E. a three-sided polygon F. an angle that measures greater than 0 and less than 90 Measure Angles Use a protractor to measure each angle Use a protractor to draw an angle with each of the following measures Solve Equations with Fractions Solve _ 2 x + 7 = x - _ 2 3 = _ x - _ 1 = _ y = 5y -_ Connect Words and Algebra Write an equation for each statement. 16. Tanya s age t is three times Martin s age m. 17. Twice the length of a segment x is 9 ft. 18. The sum of 53 and twice an angle measure y is The price of a radio r is $25 less than the price of a CD player p. 20. Half the amount of liquid j in a jar is 5 oz more than the amount of liquid b in a bowl. Triangle Congruence 213

3 The information below unpacks the standards. The Academic Vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard 4.0 Students prove basic theorems involving congruence and similarity. (Lessons 4-5, 4-8) involving relating to Academic Vocabulary Chapter Concept You learn how to use the ASA, AAS, and HL theorems to prove triangles congruent, construct triangles, and solve problems. You also learn theorems about isosceles and equilateral triangles. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. (Lessons 4-3, 4-4, 4-5, 4-6, Extension) 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. (Lessons 4-1, 4-2, 4-8) 13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. (Lesson 4-2) 17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. (Lessons 4-7, 4-8) concept idea corresponding matching interior inside exterior outside relationships links coordinate geometry a form of geometry that uses a set of numbers to describe the exact position of a figure with reference to the x- and y-axes You learn how a triangle congruence statement indicates corresponding parts. You also learn how to use the SSS and SAS theorems to prove triangles congruent and to prove that constructions are valid. You classify triangles by their angle measures and side lengths. You find the measures of interior and exterior angles of triangles and use theorems relating to these angles. You learn how angles are related and apply the Triangle Sum Theorem, the Exterior Angle Theorem, and the Third Angles Theorem. You learn how to position figures in the coordinate plane for use in coordinate proofs. Standards 1.0, 2.0, 3.0, and 16.0 are also covered in this chapter. To see these standards unpacked, go to Chapter 1, p. 4 and Chapter 2, p Chapter 4

4 Reading Strategy: Read Geometry Symbols In Geometry we often use symbols to communicate information. When studying each lesson, read both the symbols and the words slowly and carefully. Reading aloud can sometimes help you translate symbols into words. Throughout this course, you will use these symbols and combinations of these symbols to represent various geometric statements. Symbol Combinations Translated into Words ST UV Line ST is parallel to line UV. BC GH Segment BC is perpendicular to segment GH. p q If p, then q. m QRS = 45 CDE LMN The measure of angle QRS is 45 degrees. Angle CDE is congruent to angle LMN. Try This Rewrite each statement using symbols. 1. the absolute value of 2 times pi 2. The measure of angle 2 is 125 degrees. 3. Segment XY is perpendicular to line BC. 4. If not p, then not q. Translate the symbols into words. 5. m FGH = m VWX 6. ZA TU 7. p q 8. ST bisects TSU. Triangle Congruence 215

5 4-1 Classifying Triangles Objectives Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Vocabulary acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Who uses this? Manufacturers use properties of triangles to calculate the amount of material needed to make triangular objects. (See Example 4.) A triangle is a steel percussion instrument in the shape of an equilateral triangle. Different-sized triangles produce different musical notes when struck with a metal rod. Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. AB, BC, and AC are the sides of ABC. A, B, and C are the triangle s vertices. Triangle Classification By Angle Measures Acute Triangle Equiangular Triangle Right Triangle Obtuse Triangle Three acute angles Three congruent acute angles One right angle One obtuse angle EXAMPLE 1 Classifying Triangles by Angle Measures California Standards 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Classify each triangle by its angle measures. A EHG EHG is a right angle. So EHG is a right triangle. B EFH EFH and HFG form a linear pair, so they are supplementary. Therefore m EFH + m HFG = 180. By substitution, m EFH + 60 = 180. So m EFH = 120. EFH is an obtuse triangle by definition. 1. Use the diagram to classify FHG by its angle measures. 216 Chapter 4 Triangle Congruence

6 Triangle Classification By Side Lengths Equilateral Triangle Isosceles Triangle Scalene Triangle Three congruent sides At least two congruent sides No congruent sides EXAMPLE 2 Classifying Triangles by Side Lengths When you look at a figure, you cannot assume segments are congruent based on their appearance. They must be marked as congruent. Classify each triangle by its side lengths. A ABC From the figure, AB AC. So AC = 15, and ABC is equilateral. B ABD By the Segment Addition Postulate, BD = BC + CD = = 20. Since no sides are congruent, ABD is scalene. 2. Use the diagram to classify ACD by its side lengths. EXAMPLE 3 Using Triangle Classification Find the side lengths of the triangle. Step 1 Find the value of x. JK KL JK = KL (4x - 1.3) = (x + 3.2) 3x = 4.5 x = 1.5 Given Def. of segs. Substitute (4x - 13) for JK and (x + 3.2) for KL. Add 1.3 and subtract x from both sides. Divide both sides by 3. Step 2 Substitute 1.5 into the expressions to find the side lengths. JK = 4x = 4 (1.5) = 4.7 KL = x = (1.5) = 4.7 JL = 5x = 5 (1.5) = Find the side lengths of equilateral FGH. 4-1 Classifying Triangles 217

7 EXAMPLE 4 Music Application A manufacturer produces musical triangles by bending pieces of steel into the shape of an equilateral triangle. The triangles are available in side lengths of 4 inches, 7 inches, and 10 inches. How many 4-inch triangles can the manufacturer produce from a 100 inch piece of steel? The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3 (4) = 12 in. To find the number of triangles that can be made from 100 inches. of steel, divide 100 by the amount of steel needed for one triangle = 8 _ 1 3 triangles There is not enough steel to complete a ninth triangle. So the manufacturer can make 8 triangles from a 100 in. piece of steel. Each measure is the side length of an equilateral triangle. Determine how many triangles can be formed from a 100 in. piece of steel. 4a. 7 in. 4b. 10 in. THINK AND DISCUSS 1. For DEF, name the three pairs of consecutive sides and the vertex formed by each. 2. Sketch an example of an obtuse isosceles triangle, or explain why it is not possible to do so. 3. Is every acute triangle equiangular? Explain and support your answer with a sketch. 4. Use the Pythagorean Theorem to explain why you cannot draw an equilateral right triangle. 5. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe each type of triangle. 218 Chapter 4 Triangle Congruence

8 California Standards 4-1 Exercises 3.0, 12.0, 16.0, 7AF4.1, 7MG3.3, 1A2.0 SEE EXAMPLE 1 p. 216 SEE EXAMPLE 2 p. 217 SEE EXAMPLE 3 p. 217 KEYWORD: MG7 4-1 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. In JKL, JK, KL, and JL are equal. How does this help you classify JKL by its side lengths? 2. XYZ is an obtuse triangle. What can you say about the types of angles in XYZ? Classify each triangle by its angle measures. 3. DBC 4. ABD 5. ADC Classify each triangle by its side lengths. 6. EGH 7. EFH 8. HFG Multi-Step Find the side lengths of each triangle SEE EXAMPLE 4 p Crafts A jeweler creates triangular earrings by bending pieces of silver wire. Each earring is an isosceles triangle with the dimensions shown. How many earrings can be made from a piece of wire that is 50 cm long? Independent Practice For See Exercises Example Extra Practice Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Classify each triangle by its angle measures. 12. BEA 13. DBC 14. ABC Classify each triangle by its side lengths. 15. PST 16. RSP 17. RPT Multi-Step Find the side lengths of each triangle Draw a triangle large enough to measure. Label the vertices X, Y, and Z. a. Name the three sides and three angles of the triangle. b. Use a ruler and protractor to classify the triangle by its side lengths and angle measures. 4-1 Classifying Triangles 219

9 Carpentry Use the following information for Exercises 21 and 22. A manufacturer makes trusses, or triangular supports, for the roofs of houses. Each truss is the shape of an isosceles triangle in which PQ PR. The length of the base QR is 4 the length of each of the congruent sides The perimeter of each truss is 60 ft. Find each side length. 22. How many trusses can the manufacturer make from 150 feet of lumber? Draw an example of each type of triangle or explain why it is not possible. 23. isosceles right 24. equiangular obtuse 25. scalene right 26. equilateral acute 27. scalene equiangular 28. isosceles acute 29. An equilateral triangle has a perimeter of 105 in. What is the length of each side of the triangle? Architecture Classify each triangle by its angles and sides. 30. ABC 31. ACD 32. An isosceles triangle has a perimeter of 34 cm. The congruent sides measure (4x - 1) cm. The length of the third side is x cm. What is the value of x? Daniel Burnham designed and built the 22-story Flatiron Building in New York City in Source: Architecture The base of the Flatiron Building is a triangle bordered by three streets: Broadway, Fifth Avenue, and East Twenty-second Street. The Fifth Avenue side is 1 ft shorter than twice the East Twenty-second Street side. The East Twenty-second Street side is 8 ft shorter than half the Broadway side. The Broadway side is 190 ft. a. Find the two unknown side lengths. b. Classify the triangle by its side lengths. 34. Critical Thinking Is every isosceles triangle equilateral? Is every equilateral triangle isosceles? Explain. Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 35. An acute triangle is a scalene triangle. 36. A scalene triangle is an obtuse triangle. 37. An equiangular triangle is an isosceles triangle. 38. Write About It Write a formula for the side length s of an equilateral triangle, given the perimeter P. Explain how you derived the formula. 39. Construction Use the method for constructing congruent segments to construct an equilateral triangle. 40. This problem will prepare you for the Concept Connection on page 238. Marc folded a rectangular sheet of paper, ABCD, in half along EF. He folded the resulting square diagonally and then unfolded the paper to create the creases shown. a. Use the Pythagorean Theorem to find DE and CE. b. What is the m DEC? c. Classify DEC by its side lengths and by its angle measures. 220 Chapter 4 Triangle Congruence

10 41. What is the side length of an equilateral triangle with a perimeter of inches? 36 _ 2 3 inches 12 _ 1 3 inches 18 1 _ 3 inches 12 2 _ 9 inches 42. The vertices of RST are R (3, 2), S (-2, 3), and T (-2, 1). Which of these best describes RST? Isosceles Scalene Equilateral Right 43. Which of the following is NOT a correct classification of LMN? Acute Equiangular Isosceles Right 44. Gridded Response ABC is isosceles, and AB AC. AB = ( 1 BC = ( x ). What is the perimeter of ABC? 2 x + 1 4), and CHALLENGE AND EXTEND 45. A triangle has vertices with coordinates (0, 0), (a, 0), and (0, a), where a 0. Classify the triangle in two different ways. Explain your answer. 46. Write a two-column proof. Given: ABC is equiangular. EF AC Prove: EFB is equiangular. 47. Two sides of an equilateral triangle measure (y + 10) units and ( y 2-2) units. If the perimeter of the triangle is 21 units, what is the value of y? 48. Multi-Step The average length of the sides of PQR is 24. How much longer then the average is the longest side? SPIRAL REVIEW Name the parent function of each function. (Previous course) 49. y = 5x y = 3x y = 2 (x - 8) Determine if each biconditional is true. If false, give a counter example. (Lesson 2-4) 52. Two lines are parallel if and only if they do not intersect. 53. A triangle is equiangular if and only if it has three congruent angles. 54. A number is a multiple of 20 if and only if the number ends in a 0. Determine whether each line is parallel to, is perpendicular to, or coincides with y = 4x. (Lesson 3-6) 55. y = 4x y = -x _ 2 y = 2x y = _ 1 2 x 4-1 Classifying Triangles 221

11 4-2 Use with Lesson 4-2 Activity Develop the Triangle Sum Theorem In this lab, you will use patty paper to discover a relationship between the measures of the interior angles of a triangle. California Standards 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 1 Draw and label ABC on a sheet of notebook paper. 2 On patty paper draw a line l and label a point P on the line. 3 Place the patty paper on top of the triangle you drew. Align the papers so that AB is on line l and P and B coincide. Trace B. Rotate the triangle and trace C adjacent to B. Rotate the triangle again and trace A adjacent to C. The diagram shows your final step. Try This 1. What do you notice about the three angles of the triangle that you traced? 2. Repeat the activity two more times using two different triangles. Do you get the same results each time? 3. Write an equation describing the relationship among the measures of the angles of ABC. 4. Use inductive reasoning to write a conjecture about the sum of the measures of the angles of a triangle. 222 Chapter 4 Triangle Congruence

12 4-2 Angle Relationships in Triangles Objectives Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Vocabulary auxiliary line corollary interior exterior interior angle exterior angle remote interior angle Who uses this? Surveyors use triangles to make measurements and create boundaries. (See Example 1.) Triangulation is a method used in surveying. Land is divided into adjacent triangles. By measuring the sides and angles of one triangle and applying properties of triangles, surveyors can gather information about adjacent triangles. This engraving shows the county surveyor and commissioners laying out the town of Baltimore in Theorem Triangle Sum Theorem The sum of the angle measures of a triangle is 180. m A + m B + m C = 180 The proof of the Triangle Sum Theorem uses an auxiliary line. An auxiliary line is a line that is added to a figure to aid in a proof. California Standards 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles. Also covered: 2.0 PROOF Triangle Sum Theorem Given: ABC Prove: m 1 + m 2 + m 3 = 180 Proof: 4-2 Angle Relationships in Triangles 223

13 EXAMPLE 1 Surveying Application The map of France commonly used in the 1600s was significantly revised as a result of a triangulation land survey. The diagram shows part of the survey map. Use the diagram to find the indicated angle measures A m NKM m KMN + m MNK + m NKM = m NKM = m NKM = 180 m NKM = 44 Sum Thm. Substitute 88 for m KMN and 48 for m MNK. Simplify. Subtract 136 from both sides. B m JLK Step 1 Find m JKL. m NKM + m MKJ + m JKL = m JKL = m JKL = 180 m JKL = 32 Lin. Pair Thm. & Add. Post. Substitute 44 for m NKM and 104 for m MKJ. Simplify. Subtract 148 from both sides. Step 2 Use substitution and then solve for m JLK. m JLK + m JKL + m KJL = 180 Sum Thm. m JLK = 180 Substitute 32 for m JKL and 70 for m KJL. m JLK = 180 Simplify. m JLK = 78 Subtract 102 from both sides. 1. Use the diagram to find m MJK. A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. Corollaries COROLLARY HYPOTHESIS CONCLUSION The acute angles of a right triangle are complementary. D and E are complementary. m D + m E = The measure of each angle of an equiangular triangle is 60. m A = m B = m C = 60 You will prove Corollaries and in Exercises 24 and Chapter 4 Triangle Congruence

14 EXAMPLE 2 Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures What is the measure of the other acute angle? Let the acute angles be M and N, with m M = m M + m N = 90 Acute of rt. are comp m N = 90 Substitute 22.9 for m M. m N = 67.1 Subtract 22.9 from both sides. The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? 2a b. x 2c. 48 _ 2 5 The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side. Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. 4 is an exterior angle. Its remote interior angles are 1 and 2. Theorem Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. m 4 = m 1 + m 2 You will prove Theorem in Exercise 28. EXAMPLE 3 Applying the Exterior Angle Theorem Find m J. m J + m H = m FGH 5x x - 1 = 126 Ext. Thm. Substitute 5x + 17 for m J, 6x - 1 for m H, and 126 for m FGH. 11x + 16 = 126 Simplify. 11x = 110 Subtract 16 from both sides. x = 10 Divide both sides by 11. m J = 5x + 17 = 5 (10) + 17 = Find m ACD. 4-2 Angle Relationships in Triangles 225

15 Theorem Third Angles Theorem THEOREM HYPOTHESIS CONCLUSION If two angles of one triangle are congruent to two angles of another triangle, then N T the third pair of angles are congruent. You will prove Theorem in Exercise 27. EXAMPLE 4 Applying the Third Angles Theorem You can use substitution to verify that m F = 36. m F = ( ) = 36. Find m C and m F. C F Third Thm. m C = m F Def. of. y 2 = 3y 2-72 Substitute y 2 for m C and 3 y 2-72 for m F. -2y 2 =-72 Subtract 3 y 2 from both sides. y 2 = 36 Divide both sides by -2. So m C = 36. Since m F = m C, m F = Find m P and m T. THINK AND DISCUSS 1. Use the Triangle Sum Theorem to explain why the supplement of one of the angles of a triangle equals in measure the sum of the other two angles of the triangle. Support your answer with a sketch. 2. Sketch a triangle and draw all of its exterior angles. How many exterior angles are there at each vertex of the triangle? How many total exterior angles does the triangle have? 3. GET ORGANIZED Copy and complete the graphic organizer. In each box, write each theorem in words and then draw a diagram to represent it. 226 Chapter 4 Triangle Congruence

16 4-2 California Standards Exercises 2.0, 12.0, 13.0, KEYWORD: MG NS1.2, 7AF1.0, 1A2.0, 1A5.0 7AF4.1, KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. To remember the meaning of remote interior angle, think of a television remote control. What is another way to remember the term remote? 2. An exterior angle is drawn at vertex E of DEF. What are its remote interior angles? 3. What do you call segments, rays, or lines that are added to a given diagram? SEE EXAMPLE 1 p. 224 Astronomy Use the following information for Exercises 4 and 5. An asterism is a group of stars that is easier to recognize than a constellation. One popular asterism is the Summer Triangle, which is composed of the stars Deneb, Altair, and Vega. 4. What is the value of y? 5. What is the measure of each angle in the Summer Triangle? SEE EXAMPLE 2 p. 225 SEE EXAMPLE 3 p. 225 The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? y _ 2 3 Find each angle measure. 9. m M 10. m L 11. In ABC, m A = 65, and the measure of an exterior angle at C is 117. Find m B and the m BCA. SEE EXAMPLE 4 p m C and m F 13. m S and m U 14. For ABC and XYZ, m A = m X and m B = m Y. Find the measures of C and Z if m C = 4x + 7 and m Z = 3 (x + 5). 4-2 Angle Relationships in Triangles 227

17 Independent Practice For See Exercises Example Extra Practice Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 15. Navigation A sailor on ship A measures the angle between ship B and the pier and finds that it is 39. A sailor on ship B measures the angle between ship A and the pier and finds that it is 57. What is the measure of the angle between ships A and B? Ship A The measure of one of the acute angles in a right triangle is given. What is the measure of the other acute angle? _ x º Pier Ship B 57º Find each angle measure. 19. m XYZ 20. m C 21. m N and m P 22. m Q and m S 23. Multi-Step The measures of the angles of a triangle are in the ratio 1 : 4 : 7. What are the measures of the angles? (Hint: Let x, 4x, and 7x represent the angle measures.) 24. Complete the proof of Corollary Given: DEF with right F Prove: D and E are complementary. Proof: Statements 1. DEF with rt. F 2. b.? 3. m D + m E + m F = m D + m E + 90 = e.? 6. D and E are comp. Reasons 1. a.? 2. Def. of rt. 3. c.? 4. d.? 5. Subtr. Prop. 6. f.? 25. Prove Corollary using two different methods of proof. Given: ABC is equiangular. Prove: m A = m B = m C = Multi-Step The measure of one acute angle in a right triangle is 1 1 times the 4 measure of the other acute angle. What is the measure of the larger acute angle? 27. Write a two-column proof of the Third Angles Theorem. 228 Chapter 4 Triangle Congruence

18 28. Prove the Exterior Angle Theorem. Given: ABC with exterior angle ACD Prove: m ACD = m A + m B (Hint: BCA and DCA form a linear pair.) Find each angle measure. 29. UXW 30. UWY 31. WZX 32. XYZ 33. Critical Thinking What is the measure of any exterior angle of an equiangular triangle? What is the sum of the exterior angle measures? 34. Find m SRQ, given that P U, Q T, and m RST = Multi-Step In a right triangle, one acute angle measure is 4 times the other acute angle measure. What is the measure of the smaller angle? 36. Aviation To study the forces of lift and drag, the Wright brothers built a glider, attached two ropes to it, and flew it like a kite. They modeled Drag the two wind forces as the legs of a right triangle. a. What part of a right triangle is formed by Lift xº each rope? yº Rope b. Use the Triangle Sum Theorem to write an equation relating the angle measures in the right triangle. zº c. Simplify the equation from part b. What is the relationship between x and y? d. Use the Exterior Angle Theorem to write an expression for z in terms of x. e. If x = 37, use your results from parts c and d to find y and z. 37. Estimation Draw a triangle and two exterior angles at each vertex. Estimate the measure of each angle. How are the exterior angles at each vertex related? Explain. 38. Given: AB BD, BD DC, A C Prove: AD CB 39. Write About It A triangle has angle measures of 115, 40, and 25. Explain how to find the measures of the triangle s exterior angles. Support your answer with a sketch. 40. This problem will prepare you for the Concept Connection on page 238. One of the steps in making an origami crane involves folding a square sheet of paper into the shape shown. a. DCE is a right angle. FC bisects DCE, and BC bisects FCE. Find m FCB. b. Use the Triangle Sum Theorem to find m CBE. 4-2 Angle Relationships in Triangles 229

19 41. What is the value of x? Find the value of s A and B are the remote interior angles of BCD in ABC. Which of these equations must be true? m A = m B m A = 90 - m B m BCD = m BCA - m A m B = m BCD - m A 44. Extended Response The measures of the angles in a triangle are in the ratio 2 : 3 : 4. Describe how to use algebra to find the measures of these angles. Then find the measure of each angle and classify the triangle. CHALLENGE AND EXTEND 45. An exterior angle of a triangle measures 117. Its remote interior angles measure ( 2y 2 + 7) and (61 - y 2 ). Find the value of y. 46. Two parallel lines are intersected by a transversal. What type of triangle is formed by the intersection of the angle bisectors of two same-side interior angles? Explain. (Hint: Use geometry software or construct a diagram of the angle bisectors of two same-side interior angles.) 47. Critical Thinking Explain why an exterior angle of a triangle cannot be congruent to a remote interior angle. 48. Probability The measure of each angle in a triangle is a multiple of 30. What is the probability that the triangle has at least two congruent angles? 49. In ABC, m B is 5 less than 1 1 times m A. m C is 5 less than 2 1 times m A. 2 2 What is m A in degrees? SPIRAL REVIEW Make a table to show the value of each function when x is -2, 0, 1, and 4. (Previous course) 50. f (x) = 3x f (x) = x f (x) = (x - 3) Find the length of NQ. Name the theorem or postulate that justifies your answer. (Lesson 1-2) Classify each triangle by its side lengths. (Lesson 4-1) 54. ACD 55. BCD 56. ABD 57. What if? If CA = 8, What is the effect on the classification of ACD? 230 Chapter 4 Triangle Congruence

20 4-3 Congruent Triangles Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. Vocabulary corresponding angles corresponding sides congruent polygons California Standards 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Also covered: 2.0 Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices. Who uses this? Machinists used triangles to construct a model of the International Space Station s support structure. Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding angles and sides are congruent. Thus triangles that are the same size and shape are congruent. Properties of Congruent Polygons DIAGRAM ABC DEF polygon PQRS polygon WXYZ CORRESPONDING ANGLES A D B E C F P W Q X R Y S Z CORRESPONDING SIDES AB DE BC EF AC DF PQ WX QR XY RS YZ PS WZ To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts. EXAMPLE 1 Naming Congruent Corresponding Parts RST and XYZ represent the triangles of the space station s support structure. If RST XYZ, identify all pairs of congruent corresponding parts. Angles: R X, S Y, T Z Sides: RS XY, ST YZ, RT XZ 1. If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. 4-3 Congruent Triangles 231

21 EXAMPLE 2 Using Corresponding Parts of Congruent Triangles When you write a statement such as ABC DEF, you are also stating which parts are congruent. Given: EFH GFH A Find the value of x. FHE and FHG are rt.. FHE FHG m FHE = m FHG (6x - 12) = 90 6x = 102 x = 17 B Find m GFH. m EFH + m FHE + m E = 180 m EFH = 180 Def. of lines Rt. Thm. Def. of m EFH = 180 m EFH = 68.4 GFH EFH m GFH = m EFH m GFH = 68.4 Substitute values for m FHE and m FHG. Add 12 to both sides. Divide both sides by 6. Sum Thm. Substitute values for m FHE and m E. Simplify. Subtract from both sides. Corr. of are. Def. of Trans. Prop. of = Given: ABC DEF 2a. Find the value of x. 2b. Find m F. EXAMPLE 3 Proving Triangles Congruent Given: P and M are right angles. R is the midpoint of PM. PQ MN, QR NR Prove: PQR MNR Proof: Statements Reasons 1. P and M are rt. 2. P M 3. PRQ MRN 4. Q N 5. R is the mdpt. of PM. 6. PR MR 7. PQ MN ; QR NR 8. PQR MNR 1. Given 2. Rt. Thm. 3. Vert. Thm. 4. Third Thm. 5. Given 6. Def. of mdpt. 7. Given 8. Def. of 3. Given: AD bisects BE. BE bisects AD. AB DE, A D Prove: ABC DEC 232 Chapter 4 Triangle Congruence

22 Overlapping Triangles With overlapping triangles, it helps me to redraw the triangles separately. That way I can mark what I know about one triangle without getting confused by the other one. Cecelia Medina Lamar High School EXAMPLE 4 Engineering Application The bars that give structural support to a roller coaster form triangles. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. Given: JK KL, ML KL, KLJ LKM, JK ML, JL MK Prove: JKL MLK Proof: Statements 1. JK KL, ML KL 2. JKL and MLK are rt.. 3. JKL MLK 4. KLJ LKM 5. KJL LMK 6. JK ML, JL MK 7. KL LK 8. JKL MLK Reasons 1. Given 2. Def. of lines 3. Rt. Thm. 4. Given 5. Third Thm. 6. Given 7. Reflex. Prop. of 8. Def. of 4. Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ML, JK ML Prove: JKN LMN THINK AND DISCUSS 1. A roof truss is a triangular structure that supports a roof. How can you be sure that two roof trusses are the same size and shape? 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, name the congruent corresponding parts. 4-3 Congruent Triangles 233

23 4-3 California Standards Exercises 2.0, 5.0, 12.0, 7AF1.0, KEYWORD: MG AF4.1, 7MG3.4, 1A2.0, 1A5.0 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. An everyday meaning of corresponding is matching. How can this help you find the corresponding parts of two triangles? 2. If ABC RST, what angle corresponds to S? SEE EXAMPLE 1 p. 231 Given: RST LMN. Identify the congruent corresponding parts. 3. RS? 4. LN? 5. S? 6. TS? 7. L? 8. N? SEE EXAMPLE 2 p. 232 SEE EXAMPLE 3 p. 232 Given: FGH JKL. Find each value. 9. KL 10. x 11. Given: E is the midpoint of AC and BD. AB CD, AB CD Prove: ABE CDE Proof: 1. AB CD Statements 2. ABE CDE, BAE DCE 3. AB CD 4. E is the mdpt. of AC and BD. 5. e.? 6. AEB CED 7. ABE CDE 1. a.? Reasons 2. b.? 3. c.? 4. d.? 5. Def. of mdpt. 6. f.? 7. g.? SEE EXAMPLE 4 p Engineering The geodesic dome shown is a 14-story building that models Earth. Use the given information to prove that the triangles that make up the sphere are congruent. Given: SU ST SR, TU TR, UST RST, and U R Prove: RTS UTS U S T R 234 Chapter 4 Triangle Congruence

24 Independent Practice For See Exercises Example Extra Practice Skills Practice p. S10 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Given: Polygon CDEF polygon KLMN. Identify the congruent corresponding parts. 13. DE? 14. KN? 15. F? 16. L? Given: ABD CBD. Find each value. 17. m C 18. y 19. Given: MP bisects NMR. P is the midpoint of NR. MN MR, N R Prove: MNP MRP Proof: Statements Reasons 1. N R 2. MP bisects NMR. 3. c.? 4. d.? 5. P is the mdpt. of NR. 6. f.? 7. MN MR 8. MP MP 9. MNP MRP 1. a.? 2. b.? 3. Def. of bisector 4. Third Thm. 5. e.? 6. Def. of mdpt. 7. g.? 8. h.? 9. Def. of 20. Hobbies In a garden, triangular flower A beds are separated by straight rows of grass as shown. Given: ADC and BCD are right angles. AC BD, AD BC DAC CBD Prove: ADC BCD D 21. For two triangles, the following corresponding parts are given: GS KP, GR KH, SR PH, S P, G K, and R H. Write three different congruence statements. 22. The two polygons in the diagram are congruent. Complete the following congruence statement for the polygons. polygon R? polygon V? Write and solve an equation for each of the following. 23. ABC DEF. AB = 2x - 10, and DE = x Find the value of x and AB. 24. JKL MNP. m L = ( x ), and m P = ( 2x 2 + 1). What is m L? 25. Polygon ABCD polygon PQRS. BC = 6x + 5, and QR = 5x + 7. Find the value of x and BC. E B C 4-3 Congruent Triangles 235

25 26. This problem will prepare you for the Concept Connection on page 238. Many origami models begin with a square piece of paper, JKLM, that is folded along both diagonals to make the creases shown. JL and MK are perpendicular bisectors of each other, and NML NKL. a. Explain how you know that KL and ML are congruent. b. Prove NML NKL. 27. Draw a diagram and then write a proof. Given: BD AC. D is the midpoint of AC. AB CB, and BD bisects ABC. Prove: ABD CBD 28. Critical Thinking Draw two triangles that are not congruent but have an area of 4 cm 2 each. 29. /ERROR ANALYSIS / Given MPQ EDF. Two solutions for finding m E are shown. Which is incorrect? Explain the error. 30. Write About It Given the diagram of the triangles, is there enough information to prove that HKL is congruent to YWX? Explain. 31. Which congruence statement correctly indicates that the two given triangles are congruent? ABC EFD ABC DEF ABC FDE ABC FED 32. MNP RST. What are the values of x and y? x = 26, y = 21 _ 1 x = 25, y = 20 _ x = 27, y = 20 x = 30 _ 1 3, y = 16 _ ABC XYZ. m A = 47.1, and m C = Find m Y MNR SPQ, NL = 18, SP = 33, SR = 10, RQ = 24, and QP = 30. What is the perimeter of MNR? Chapter 4 Triangle Congruence

26 CHALLENGE AND EXTEND 35. Multi-Step Given that the perimeter of TUVW is 149 units, find the value of x. Is TUV TWV? Explain. 36. Multi-Step Polygon ABCD polygon EFGH. A is a right angle. m E = ( y 2-10), and m H = ( 2y 2-132). Find m D. 37. Given: RS RT, S T Prove: RST RTS SPIRAL REVIEW Two number cubes are rolled. Find the probability of each outcome. (Previous course) 38. Both numbers rolled are even. 39. The sum of the numbers rolled is 5. Classify each angle by its measure. (Lesson 1-3) 40. m DOC = m BOA = m COA = 140 Find each angle measure. (Lesson 4-2) 43. Q 44. P 45. QRS Q: What math classes did you take in high school? A: Algebra 1 and 2, Geometry, Precalculus KEYWORD: MG7 Career Q: What kind of degree or certification will you receive? A: I will receive an associate s degree in applied science. Then I will take an exam to be certified as an EMT or paramedic. Q: How do you use math in your hands-on training? A: I calculate dosages based on body weight and age. I also calculate drug doses in milligrams per kilogram per hour or set up an IV drip to deliver medications at the correct rate. Jordan Carter Emergency Medical Services Program Q: What are your future career plans? A: When I am certified, I can work for a private ambulance service or with a fire department. I could also work in a hospital, transporting critically ill patients by ambulance or helicopter. 4-3 Congruent Triangles 237

27 SECTION 4A Triangles and Congruence Origami Origami is the Japanese art of paper folding. The Japanese word origami literally means fold paper. This ancient art form relies on properties of geometry to produce fascinating and beautiful shapes. Each of the figures shows a step in making an origami swan from a square piece of paper. The final figure shows the creases of an origami swan that has been unfolded. Step 1 Step 2 Step 3 Fold the paper in half diagonally and crease it. Turn it over. Fold corners A and C to the center line and crease. Turn it over. Fold in half along the center crease so that DE and DF are together. Step 4 Step 5 Step 6 Fold the narrow point upward at a 90 angle and crease. Push in the fold so that the neck is inside the body. Fold the tip downward and crease. Push in the fold so that the head is inside the neck. Fold up the flap to form the wing. 1. Use the fact that ABCD is a square to classify ABD by its side lengths and by its angle measures. 2. DB bisects ABC and ADC. DE bisects ADB. Find the measures of the angles in EDB. Explain how you found the measures. 3. Given that DB bisects ABC and EDF, BE BF, and DE DF, prove that EDB FDB. 238 Chapter 4 Triangle Congruence

28 Quiz for Lessons 4-1 Through 4-3 SECTION 4A 4-1 Classifying Triangles Classify each triangle by its angle measures. 1. ACD 2. ABD 3. ADE Classify each triangle by its side lengths. 4. PQR 5. PRS 6. PQS 4-2 Angle Relationships in Triangles Find each angle measure. 7. m M 8. m ABC 9. A carpenter built a triangular support structure for a roof. Two of the angles of the structure measure 37 and 55. Find the measure of RTP, the angle formed by the roof of the house and the roof of the patio. 4-3 Congruent Triangles Given: JKL DEF. Identify the congruent corresponding parts. 10. KL? 11. DF? 12. K? 13. F? Given: PQR STU. Find each value. 14. PQ 15. y 16. Given: AB CD, AB CD, AC BD, AC CD, DB AB Prove: ACD DBA Proof: Statements Reasons 1. AB CD 2. BAD CDA 3. AC CD, DB AB 4. ACD and DBA are rt. 5. e.? 6. f.? 7. AB CD, AC BD 8. h.? 9. ACD DBA 1. a.? 2. b.? 3. c.? 4. d.? 5. Rt. Thm. 6. Third Thm. 7. g.? 8. Reflex Prop. of 9. i.? Ready to Go On? 239

29 4-4 Use with Lesson 4-4 Activity 1 Explore SSS and SAS Triangle Congruence In Lesson 4-3, you used the definition of congruent triangles to prove triangles congruent. To use the definition, you need to prove that all three pairs of corresponding sides and all three pairs of corresponding angles are congruent. In this lab, you will discover some shortcuts for proving triangles congruent. California Standards 1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning. 1 Measure and cut six pieces from the straws: two that are 2 inches long, two that are 4 inches long, and two that are 5 inches long. 2 Cut two pieces of string that are each about 20 inches long. 3 Thread one piece of each size of straw onto a piece of string. Tie the ends of the string together so that the pieces of straw form a triangle. 4 Using the remaining pieces, try to make another triangle with the same side lengths that is not congruent to the first triangle. Try This 1. Repeat Activity 1 using side lengths of your choice. Are your results the same? 2. Do you think it is possible to make two triangles that have the same side lengths but that are not congruent? Why or why not? 3. How does your answer to Problem 2 provide a shortcut for proving triangles congruent? 4. Complete the following conjecture based on your results. Two triangles are congruent if?. 240 Chapter 4 Triangle Congruence

30 Activity 2 1 Measure and cut two pieces from the straws: one that is 4 inches long and one that is 5 inches long. 2 Use a protractor to help you bend a paper clip to form a 30 angle. 3 Place the pieces of straw on the sides of the 30 angle. The straws will form two sides of your triangle. 4 Without changing the angle formed by the paper clip, use a piece of straw to make a third side for your triangle, cutting it to fit as necessary. Use additional paper clips or string to hold the straws together in a triangle. Try This 5. Repeat Activity 2 using side lengths and an angle measure of your choice. Are your results the same? 6. Suppose you know two side lengths of a triangle and the measure of the angle between these sides. Can the length of the third side be any measure? Explain. 7. How does your answer to Problem 6 provide a shortcut for proving triangles congruent? 8. Use the two given sides and the given angle from Activity 2 to form a triangle that is not congruent to the triangle you formed. (Hint: One of the given sides does not have to be adjacent to the given angle.) 9. Complete the following conjecture based on your results. Two triangles are congruent if?. 4-4 Geometry Lab 241

31 4-4 Triangle Congruence: SSS and SAS Objectives Apply SSS and SAS to construct triangles and to solve problems. Prove triangles congruent by using SSS and SAS. Vocabulary triangle rigidity included angle California Standards 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Also covered: 2.0, 16.0 Who uses this? Engineers used the property of triangle rigidity to design the internal support for the Statue of Liberty and to build bridges, towers, and other structures. (See Example 2.) In Lesson 4-3, you proved triangles congruent by showing that all six pairs of corresponding parts were congruent. The property of triangle rigidity gives you a shortcut for proving two triangles congruent. It states that if the side lengths of a triangle are given, the triangle can have only one shape. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate. Postulate Side-Side-Side (SSS) Congruence POSTULATE HYPOTHESIS CONCLUSION If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. ABC FDE EXAMPLE 1 Using SSS to Prove Triangle Congruence Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent parts. Use SSS to explain why PQR PSR. It is given that PQ PS and that QR SR. By the Reflexive Property of Congruence, PR PR. Therefore PQR PSR by SSS. 1. Use SSS to explain why ABC CDA. An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC. 242 Chapter 4 Triangle Congruence

32 It can also be shown that only two pairs of congruent corresponding sides are needed to prove the congruence of two triangles if the included angles are also congruent. Postulate Side-Angle-Side (SAS) Congruence POSTULATE HYPOTHESIS CONCLUSION If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. ABC EFD EXAMPLE 2 Engineering Application The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides. The figure shows part of the support structure of the Statue of Liberty. Use SAS to explain why KPN LPM. K It is given that KP LP and that NP MP. By the Vertical Angles Theorem, KPN LPM. Therefore KPN LPM N by SAS. P M L 2. Use SAS to explain why ABC DBC. The SAS Postulate guarantees that if you are given the lengths of two sides and the measure of the included angle, you can construct one and only one triangle. Construction Congruent Triangles Using SAS Use a straightedge to draw two segments and one angle, or copy the given segments and angle. Construct AB congruent to one of the segments. Construct A congruent to the given angle. Construct AC congruent to the other segment. Draw CB to complete ABC. 4-4 Triangle Congruence: SSS and SAS 243

33 EXAMPLE 3 Verifying Triangle Congruence Show that the triangles are congruent for the given value of the variable. A UVW YXW, x = 3 ZY = x - 1 = 3-1 = 2 XZ = x = 3 XY = 3x - 5 = 3 (3) - 5 = 4 UV YX. VW XZ, and UW YZ. So UVW YXZ by SSS. B DEF JGH, y = 7 JG = 2y + 1 = 2 (7) + 1 = 15 GH = y 2-4y + 3 = (7) 2-4 (7) + 3 = 24 m G = 12y + 42 = 12 (7) + 42 = 126 DE JG. EF GH, and E G. So DEF JGH by SAS. 3. Show that ADB CDB when t = 4. EXAMPLE 4 Proving Triangles Congruent Given: l m, EG HF Prove: EGF HFG Proof: Statements 1. EG HF 2. l m 3. EGF HFG 4. FG GF 5. EGF HFG 1. Given 2. Given Reasons 3. Alt. Int. Thm. 4. Reflex Prop. of 5. SAS Steps 1, 3, 4 4. Given: QP bisects RQS. QR QS Prove: RQP SQP 244 Chapter 4 Triangle Congruence

34 THINK AND DISCUSS 1. Describe three ways you could prove that ABC DEF. 2. Explain why the SSS and SAS Postulates are shortcuts for proving triangles congruent. 3. GET ORGANIZED Copy and complete the graphic organizer. Use it to compare the SSS and SAS postulates. 4-4 Exercises California Standards 2.0, 5.0, 16.0, 7AF4.1, 7MG3.3, 7MG3.4, 7MR1.0, 7MR1.1, 1A2.0, 1A4.0, 1A5.0 KEYWORD: MG7 4-4 KEYWORD: MG7 Parent GUIDED PRACTICE 1. Vocabulary In RST which angle is the included angle of sides ST and TR? SEE EXAMPLE 1 p. 242 Use SSS to explain why the triangles in each pair are congruent. 2. ABD CDB 3. MNP MQP SEE EXAMPLE 2 p Sailing Signal flags are used to communicate messages when radio silence is required. The Zulu signal flag means, I require a tug. GJ = GH = GL = GK = 20 in. Use SAS to explain why JGK LGH. J G H K L SEE EXAMPLE 3 p. 244 Show that the triangles are congruent for the given value of the variable. 5. GHJ IHJ, x = 4 6. RST TUR, x = Triangle Congruence: SSS and SAS 245

35 SEE EXAMPLE 4 p Given: JK ML, JKL MLK Prove: JKL MLK Proof: Statements 1. JK ML 2. b.? 3. KL LK 4. JKL MLK Reasons 1. a.? 2. Given 3. c.? 4. d.? Independent Practice For See Exercises Example Extra Practice Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Use SSS to explain why the triangles in each pair are congruent. 8. BCD EDC 9. GJK GJL 10. Theater The lights shining on a stage appear to form two congruent right triangles. Given EC DB, use SAS to explain why ECB DBC. Show that the triangles are congruent for the given value of the variable. 11. MNP QNP, y = XYZ STU, t = Given: B is the midpoint of DC. AB DC Prove: ABD ABC Proof: Statements 1. B is the mdpt. of DC. 2. b.? 3. c.? 4. ABD and ABC are rt.. 5. ABD ABC 6. f.? 7. ABD ABC Reasons 1. a.? 2. Def. of mdpt. 3. Given 4. d.? 5. e.? 6. Reflex. Prop. of 7. g.? 246 Chapter 4 Triangle Congruence

36 Which postulate, if any, can be used to prove the triangles congruent? Explain what additional information, if any, you would need to prove ABC DEC by each postulate. a. SSS b. SAS Multi-Step Graph each triangle. Then use the Distance Formula and the SSS Postulate to determine whether the triangles are congruent. 19. QRS and TUV 20. ABC and DEF Q (-2, 0), R (1, -2), S (-3, -2) A (2, 3), B (3, -1), C (7, 2) T (5, 1), U (3, -2), V (3, 2) D (-3, 1), E (1, 2), F (-3, 5) 21. Given: ZVY WYV, ZVW WYZ, VW YZ Prove: ZVY WYV Proof: Statements Reasons 1. ZVY WYV, ZVW WYZ 2. m ZVY = m WYV, m ZVW = m WYZ 3. m ZVY + m ZVW = m WYV + m WYZ 4. c.? 5. WVY ZYV 6. VW YZ 7. f.? 8. ZVY WYV 1. a.? 2. b.? 3. Add. Prop. of = 4. Add. Post. 5. d.? 6. e.? 7. Reflex. Prop. of 8. g.? 22. This problem will prepare you for the Concept Connection on page 280. The diagram shows two triangular trusses that were built for the roof of a doghouse. a. You can use a protractor to check that A and D are right angles. Explain how you could make just two additional measurements on each truss to ensure that the trusses are congruent. b. You verify that the trusses are congruent and find that AB = AC = 2.5 ft. Find the length of EF to the nearest tenth. Explain. 4-4 Triangle Congruence: SSS and SAS 247

37 Ecology 23. Critical Thinking Draw two isosceles triangles that are not congruent but that have a perimeter of 15 cm each. 24. ABC ADC for what value of x? Explain why the SSS Postulate can be used to prove the two triangles congruent. 25. Ecology A wing deflector is a triangular structure made of logs that is filled with large rocks and placed in a stream to guide the current or prevent erosion. Wing deflectors are often used in pairs. Suppose an engineer wants to build two wing deflectors. The logs that form the sides of each wing deflector are perpendicular. How can the engineer make sure that the two wing deflectors are congruent? Wing deflectors are designed to reduce the width-to-depth ratio of a stream. Reducing the width increases the velocity of the stream. 26. Write About It If you use the same two sides and included angle to repeat the construction of a triangle, are your two constructed triangles congruent? Explain. 27. Construction Use three segments (SSS) to construct a scalene triangle. Suppose you then use the same segments in a different order to construct a second triangle. Will the result be the same? Explain. 28. Which of the three triangles below can be proven congruent by SSS or SAS? I and II II and III I and III I, II, and III 29. What is the perimeter of polygon ABCD? 29.9 cm 49.8 cm 39.8 cm 59.8 cm 30. Jacob wants to prove that FGH JKL using SAS. He knows that FG JK and FH JL. What additional piece of information does he need? F J H L G K F G 31. What must the value of x be in order to prove that EFG EHG by SSS? Chapter 4 Triangle Congruence

38 CHALLENGE AND EXTEND 32. Given:. ADC and BCD are supplementary. AD CB Prove: ADB CBD (Hint: Draw an auxiliary line.) 33. Given: QPS TPR, PQ PT, PR PS Prove: PQR PTS Algebra Use the following information for Exercises 34 and 35. Find the value of x. Then use SSS or SAS to write a paragraph proof showing that two of the triangles are congruent. 34. m FKJ = 2x 35. FJ bisects KFH. m KFJ = (3x + 10) m KFJ = (2x + 6) KJ = 4x + 8 m HFJ = (3x - 21) HJ = 6 (x - 4) FK = 8x - 45 FH = 6x + 9 SPIRAL REVIEW Solve and graph each inequality. (Previous course) 36. x_ a + 4 > 3a m Solve each equation. Write a justification for each step. (Lesson 2-5) 39. 4x - 7 = a_ + 5 = r = 4r Given: EFG GHE. Find each value. (Lesson 4-3) 42. x 43. m FEG 44. m FGH Using Technology Use geometry software to complete the following. 1. Draw a triangle and label the vertices A, B, and C. Draw a point and label it D. Mark a vector from A to B and translate D by the marked vector. Label the image E. Draw DE. Mark BAC and rotate DE about D by the marked angle. Mark ABC and rotate DE about E by the marked angle. Label the intersection F. 2. Drag A, B, and C to different locations. What do you notice about the two triangles? 3. Write a conjecture about ABC and DEF. 4. Test your conjecture by measuring the sides and angles of ABC and DEF. 4-4 Triangle Congruence: SSS and SAS 249

39 4-5 Use with Lesson 4-5 Activity 1 Predict Other Triangle Congruence Relationships Geometry software can help you investigate whether certain combinations of triangle parts will make only one triangle. If a combination makes only one triangle, then this arrangement can be used to prove two triangles congruent. California Standards 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Also covered: Construct CAB measuring 45 and EDF measuring Move EDF so that DE overlays BA. Where DF and AC intersect, label the point G. Measure DGA. 3 Move CAB to the left and right without changing the measures of the angles. Observe what happens to the size of DGA. 4 Measure the distance from A to D. Try to change the shape of the triangle without changing AD and the measures of A and D. Try This 1. Repeat Activity 1 using angle measures of your choice. Are your results the same? Explain. 2. Do the results change if one of the given angles measures 90? 3. What theorem proves that the measure of DGA in Step 2 will always be the same? 4. In Step 3 of the activity, the angle measures in ADG stayed the same as the size of the triangle changed. Does Angle-Angle-Angle, like Side-Side-Side, make only one triangle? Explain. 5. Repeat Step 4 of the activity but measure the length of AG instead of AD. Are your results the same? Does this lead to a new congruence postulate or theorem? 6. If you are given two angles of a triangle, what additional piece of information is needed so that only one triangle is made? Make a conjecture based on your findings in Step Chapter 4 Triangle Congruence

40 Activity 2 1 Construct YZ with a length of 6.5 cm. 2 Using YZ as a side, construct XYZ measuring Draw a circle at Z with a radius of 5 cm. Construct ZW, a radius of circle Z. 4 Move W around circle Z. Observe the possible shapes of YZW. Try This 7. In Step 4 of the activity, how many different triangles were possible? Does Side-Side-Angle make only one triangle? 8. Repeat Activity 2 using an angle measure of 90 in Step 2 and a circle with a radius of 7 cm in Step 3. How many different triangles are possible in Step 4? 9. Repeat the activity again using a measure of 90 in Step 2 and a circle with a radius of 8.25 cm in Step 3. Classify the resulting triangle by its angle measures. 10. Based on your results, complete the following conjecture. In a Side-Side-Angle combination, if the corresponding nonincluded angles are?, then only one triangle is possible. 4-5 Technology Lab 251

41 4-5 Triangle Congruence: ASA, AAS, and HL Objectives Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles congruent by using ASA, AAS, and HL. Vocabulary included side California Standards 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Also covered: 2.0, 4.0, 16.0 Why use this? Bearings are used to convey direction, helping people find their way to specific locations. Participants in an orienteering race use a map and a compass to find their way to checkpoints along an unfamiliar course. Directions are given by bearings, which are based on compass headings. For example, to travel along the bearing S 43 E, you face south and then turn 43 to the east. An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. PQ is the included side of P and Q. Postulate Angle-Side-Angle (ASA) Congruence POSTULATE HYPOTHESIS CONCLUSION If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. ABC DEF EXAMPLE 1 Problem-Solving Application Organizers of an orienteering race are planning a course with checkpoints A, B, and C. Does the table give enough information to determine the location of the checkpoints? 1 Understand the Problem Bearing Distance A to B N 55 E 7.6 km B to C N 26 W C to A S 20 W The answer is whether the information in the table can be used to find the position of checkpoints A, B, and C. List the important information: The bearing from A to B is N 55 E. From B to C is N 26 W, and from C to A is S 20 W. The distance from A to B is 7.6 km. 252 Chapter 4 Triangle Congruence

42 2 Make a Plan Draw the course using vertical lines to show north-south directions. Then use these parallel lines and the alternate interior angles to help find angle measures of ABC. 3 Solve m CAB = = 35 m CBA = ( ) = 99 You know the measures of CAB and CBA and the length of the included side AB. Therefore by ASA, a unique triangle ABC is determined. 4 Look Back One and only one triangle can be made using the information in the table, so the table does give enough information to determine the location of all the checkpoints. 1. What if...? If 7.6 km is the distance from B to C, is there enough information to determine the location of all the checkpoints? Explain. EXAMPLE 2 Applying ASA Congruence Determine if you can use ASA to prove UVX WVX. Explain. UXV WXV as given. Since WVX is a right angle that forms a linear pair with UVX, WVX UVX. Also VX VX by the Reflexive Property. Therefore UVX WVX by ASA. 2. Determine if you can use ASA to prove NKL LMN. Explain. Construction Congruent Triangles Using ASA Use a straightedge to draw a segment and two angles, or copy the given segment and angles. Construct CD congruent to the given segment. Construct C congruent to one of the angles. Construct D congruent to the other angle. CDE Label the intersection of the rays as E. 4-5 Triangle Congruence: ASA, AAS, and HL 253

43 You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side (AAS). Theorem Angle-Angle-Side (AAS) Congruence THEOREM HYPOTHESIS CONCLUSION If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. GHJ KLM PROOF Angle-Angle-Side Congruence Given: G K, J M, HJ LM Prove: GHJ KLM Proof: Statements 1. G K, J M 2. H L 3. HJ LM 4. GHJ KLM Reasons 1. Given 2. Third Thm. 3. Given 4. ASA Steps 1, 3, and 2 EXAMPLE 3 Using AAS to Prove Triangles Congruent Use AAS to prove the triangles congruent. Given: AB ED, BC DC Prove: ABC EDC Proof: 3. Use AAS to prove the triangles congruent. Given: JL bisects KLM. K M Prove: JKL JML There are four theorems for right triangles that are not used for acute or obtuse triangles. They are Leg-Leg (LL), Hypotenuse-Angle (HA), Leg-Angle (LA), and Hypotenuse-Leg (HL). You will prove LL, HA, and LA in Exercises 21, 23, and Chapter 4 Triangle Congruence

44 Theorem Hypotenuse-Leg (HL) Congruence THEOREM HYPOTHESIS CONCLUSION If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. ABC DEF You will prove the Hypotenuse-Leg Theorem in Lesson 4-8. EXAMPLE 4 Applying HL Congruence Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. A VWX and YXW According to the diagram, VWX and YXW are right triangles that share hypotenuse WX. WX XW by the Reflexive Property. It is given that WV XY, therefore VWX YXW by HL. B VWZ and YXZ This conclusion cannot be proved by HL. According to the diagram, VWZ and YXZ are right triangles, and WV XY. You do not know that hypotenuse WZ is congruent to hypotenuse XZ. 4. Determine if you can use the HL Congruence Theorem to prove ABC DCB. If not, tell what else you need to know. THINK AND DISCUSS 1. Could you use AAS to prove that these two triangles are congruent? Explain. 2. The arrangement of the letters in ASA matches the arrangement of what parts of congruent triangles? Include a sketch to support your answer. 3. GET ORGANIZED Copy and complete the graphic organizer. In each column, write a description of the method and then sketch two triangles, marking the appropriate congruent parts. 4-5 Triangle Congruence: ASA, AAS, and HL 255

45 4-5 California Standards Exercises 2.0, 4.0, 5.0, 12.0, KEYWORD: MG7 4-5 GUIDED PRACTICE 16.0, AF1.0, 7AF4.1, 7MG3.4 KEYWORD: MG7 Parent 1. Vocabulary A triangle contains ABC and ACB with BC closed in between them. How would this help you remember the definition of included side? SEE EXAMPLE 1 p. 252 Surveying Use the table for Exercises 2 and 3. A landscape designer surveyed the boundaries of a triangular park. She made the following table for the dimensions of the land. A A to B B to C C to A 115 ft Bearing E S 25 E N 62 W Distance 115 ft?? B 2. Draw the plot of land described by the table. Label the measures of the angles in the triangle. C 3. Does the table have enough information to determine the locations of points A, B, and C? Explain. SEE EXAMPLE 2 p. 253 Determine if you can use ASA to prove the triangles congruent. Explain. 4. VRS and VTS, given that 5. DEH and FGH VS bisects RST and RVT SEE EXAMPLE 3 p Use AAS to prove the triangles congruent. Given: R and P are right angles. QR SP Prove: QPS SRQ Proof: SEE EXAMPLE 4 p. 255 Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 7. ABC and CDA 8. XYV and ZYV 256 Chapter 4 Triangle Congruence

46 PRACTICE AND PROBLEM SOLVING Independent Practice Surveying Use the table for Exercises 9 and 10. For See From two different observation towers a fire is sighted. The locations of the towers Exercises Example are given in the following table X to Y X to F Y to F 13 3 Bearing E N 53 E N 16 W Distance 6 km?? Extra Practice Skills Practice p. S11 Application Practice p. S31 9. Draw the diagram formed by observation tower X, observation tower Y, and the fire F. Label the measures of the angles. 10. Is there enough information given in the table to pinpoint the location of the fire? Explain. Math History Euclid wrote the mathematical text The Elements around 2300 years ago. It may be the second most reprinted book in history. Determine if you can use ASA to prove the triangles congruent. Explain. 11. MKJ and MKL 12. RST and TUR 13. Given: AB DE, C F Prove: ABC DEF Proof: Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 14. GHJ and JKG 15. ABE and DCE, given that E is the midpoint of AD and BC Multi-Step For each pair of triangles write a triangle congruence statement. Identify the transformation that moves one triangle to the position of the other triangle Critical Thinking Side-Side-Angle (SSA) cannot be used to prove two triangles congruent. Draw a diagram that shows why this is true. 4-5 Triangle Congruence: ASA, AAS, and HL 257

47 19. This problem will prepare you for the Concept Connection on page 280. A carpenter built a truss to support the roof of a doghouse. a. The carpenter knows that KJ MJ. Can the carpenter conclude that KJL MJL? Why or why not? b. Suppose the carpenter also knows that JLK is a right angle. Which theorem can be used to show that KJL MJL? 20. /ERROR ANALYSIS / Two proofs that EFH GHF are given. Which is incorrect? Explain the error. 21. Write a paragraph proof of the Leg-Leg (LL) Congruence Theorem. If the legs of one right triangle are congruent to the corresponding legs of another right triangle, the triangles are congruent. 22. Use AAS to prove the triangles congruent. Given: AD BC, AD CB Prove: AED CEB Proof: Statements 1. AD BC 2. DAE BCE 3. c.? 4. d.? 5. e.? Reasons 1. a.? 2. b.? 3. Vert. Thm. 3. Given 4. f.? 23. Prove the Hypotenuse-Angle (HA) Theorem. Given: KM JL, JM LM, JMK LMK Prove: JKM LKM 24. Write About It The legs of both right DEF and right RST are 3 cm and 4 cm. They each have a hypotenuse 5 cm in length. Describe two different ways you could prove that DEF RST. 25. Construction Use the method for constructing perpendicular lines to construct a right triangle. 26. What additional congruence statement is necessary to prove XWY XVZ by ASA? XVZ XWY VZ WY VUY WUZ XZ XY 258 Chapter 4 Triangle Congruence

48 27. Which postulate or theorem justifies the congruence statement STU VUT? ASA HL SSS SAS 28. Which of the following congruence statements is true? A B CE DE AED CEB AED BEC 29. In RST, RT = 6y - 2. In UVW, UW = 2y + 7. R U, and S V. What must be the value of y in order to prove that RST UVW? Extended Response Draw a triangle. Construct a second triangle that has the same angle measures but is not congruent. Compare the lengths of each pair of corresponding sides. Consider the relationship between the lengths of the sides and the measures of the angles. Explain why Angle-Angle-Angle (AAA) is not a congruence principle. CHALLENGE AND EXTEND 31. Sports This bicycle frame includes VSU and VTU, which lie in intersecting planes. From the given angle measures, can you conclude that VSU VTU? Explain. m VUS = (7y - 2) m VUT = ( 5 1_ 2 x - _ 1 m USV = 5 _ 2 3 y m UTV = (4x + 8) m SVU = (3y - 6) m TVU = 2x 2) 32. Given: ABC is equilateral. C is the midpoint of DE. DAC and EBC are congruent and supplementary. Prove: DAC EBC 33. Write a two-column proof of the Leg-Angle (LA) Congruence Theorem. If a leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, the triangles are congruent. (Hint: There are two cases to consider.) 34. If two triangles are congruent by ASA, what theorem could you use to prove that the triangles are also congruent by AAS? Explain. SPIRAL REVIEW Identify the x- and y-intercepts. Use them to graph each line. (Previous course) 35. y = 3x y = - 1_ x y = -5x Find AB and BC if AC = 10. (Lesson 1-6) 39. Find m C. (Lesson 4-2) 4-5 Triangle Congruence: ASA, AAS, and HL 259

49 4-6 Triangle Congruence: CPCTC Objective Use CPCTC to prove parts of triangles are congruent. Vocabulary CPCTC Why learn this? You can use congruent triangles to estimate distances. CPCTC is an abbreviation for the phrase Corresponding Parts of Congruent Triangles are Congruent. It can be used as a justification in a proof after you have proven two triangles congruent. EXAMPLE 1 Engineering Application SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. To design a bridge across a canyon, you need to find the distance from A to B. Locate points C, D, and E as shown in the figure. If DE = 600 ft, what is AB? D B, because they are both right angles. DC CB,because DC = CB = 500 ft. DCE BCA, because vertical angles are congruent. Therefore DCE BCA by ASA or LA. By CPCTC, ED AB, so AB = ED = 600 ft. 1. A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? EXAMPLE 2 Proving Corresponding Parts Congruent Given: AB DC, ABC DCB Prove: A D California Standards Proof: 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Also covered: Given: PR bisects QPS and QRS. Prove: PQ PS 260 Chapter 4 Triangle Congruence

50 EXAMPLE 3 Using CPCTC in a Proof Given: EG DF, EG DF Prove: ED GF Proof: Work backward when planning a proof. To show that ED GF, look for a pair of angles that are congruent. Then look for triangles that contain these angles. Statements 1. EG DF 2. EG DF 3. EGD FDG 4. GD DG 5. EGD FDG 6. EDG FGD 7. ED GF Reasons 1. Given 2. Given 3. Alt. Int. Thm. 4. Reflex. Prop. of 5. SAS Steps 1, 3, and 4 6. CPCTC 7. Converse of Alt. Int. Thm. 3. Given: J is the midpoint of KM and NL. Prove: KL MN You can also use CPCTC when triangles are on a coordinate plane. You use the Distance Formula to find the lengths of the sides of each triangle. Then, after showing that the triangles are congruent, you can make conclusions about their corresponding parts. EXAMPLE 4 Using CPCTC in the Coordinate Plane Given: A (2, 3), B (5, -1), C (1, 0), D (-4, -1), E (0, 2), F (-1, -2) Prove: ABC DEF Step 1 Plot the points on a coordinate plane. Step 2 Use the Distance Formula to find the lengths of the sides of each triangle. D = (x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 AB = (5-2) 2 + (-1-3) 2 DE = (0 - (-4)) 2 + (2 - (-1)) 2 = = 25 = 5 = = 25 = 5 BC = (1-5) 2 + (0 - (-1)) 2 EF = (-1-0) 2 + (-2-2) 2 = = 17 = = 17 AC = (1-2) 2 + (0-3) 2 DF = (-1 - (-4)) 2 + (-2 - (-1)) 2 = = 10 = = 10 So AB DE, BC EF, and AC DF. Therefore ABC DEF by SSS, and ABC DEF by CPCTC. 4. Given: J (-1, -2), K (2, -1), L (-2, 0), R (2, 3), S (5, 2), T (1, 1) Prove: JKL RST 4-6 Triangle Congruence: CPCTC 261

51 THINK AND DISCUSS 1. In the figure, UV XY, VW YZ, and V Y. Explain why UVW XYZ. By CPCTC, which additional parts are congruent? 2. GET ORGANIZED Copy and complete the graphic organizer. Write all conclusions you can make using CPCTC. 4-6 Exercises 2.0, 5.0, 22.0, 6SDAP1.1, KEYWORD: MG7 4-6 GUIDED PRACTICE 1. Vocabulary You use CPCTC after proving triangles are congruent. Which parts of congruent triangles are referred to as corresponding parts? California Standards 7AF2.0, 7AF4.1, 7MG3.2, 7MG3.4, 7MR1.1, 1A2.0 KEYWORD: MG7 Parent SEE EXAMPLE 1 p Archaeology An archaeologist wants to find the height AB of a rock formation. She places a marker at C and steps off the distance from C to B. Then she walks the same distance from C and places a marker at D. If DE = 6.3 m, what is AB? SEE EXAMPLE 2 p Given: X is the midpoint of ST. RX ST Prove: RS RT Proof: 262 Chapter 4 Triangle Congruence

52 SEE EXAMPLE 3 p Given: AC AD, CB DB Prove: AB bisects CAD. Proof: Statements 1. AC AD, CB DB 2. b.? 3. ACB ADB 4. CAB DAB 5. AB bisects CAD Reasons 1. a.? 2. Reflex. Prop. of 3. c.? 4. d.? 5. e.? SEE EXAMPLE 4 p. 261 Multi-Step Use the given set of points to prove each congruence statement. 5. E (-3, 3), F (-1, 3), G (-2, 0), J (0, -1), K (2, -1), L (1, 2) ; EFG JKL 6. A (2, 3), B (4, 1), C (1, -1), R (-1, 0), S (-3, -2), T (0, -4) ; ACB RTS Independent Practice For See Exercises Example Extra Practice Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 7. Surveying To find the distance AB across A a river, a surveyor first locates point C. He measures the distance from C to B. 500 ft Then he locates point D the same distance B D 500 ft C east of C. If DE = 420 ft, what is AB? E 8. Given: M is the midpoint of 9. Given: WX XY YZ ZW PQ and RS. Prove: W Y Prove: QR PS 10. Given: G is the midpoint of FH. EF EH Prove: Given: LM bisects JLK. JL KL Prove: M is the midpoint of JK. Multi-Step Use the given set of points to prove each congruence statement. 12. R (0, 0), S (2, 4), T (-1, 3), U (-1, 0), V (-3, -4), W (-4, -1) ; RST UVW 13. A (-1, 1), B (2, 3), C (2, -2), D (2, -3), E (-1, -5), F (-1, 0) ; BAC EDF 14. Given: QRS is adjacent to QTS. QS bisects RQT. R T Prove: QS bisects RT. 15. Given: ABE and CDE with E the midpoint of AC and BD Prove: AB CD 4-6 Triangle Congruence: CPCTC 263

53 16. This problem will prepare you for the Concept Connection on page 280. The front of a doghouse has the dimensions shown. a. How can you prove that ADB ADC? b. Prove that BD CD. c. What is the length of BD and BC to the nearest tenth? Multi-Step Find the value of x Use the diagram for Exercises Given: PS = RQ, m 1 = m 4 Prove: m 3 = m Given: m 1 = m 2, m 3 = m 4 Prove: PS = RS 21. Given: PS = RQ, PQ = RS Prove: PQ RS 22. Critical Thinking Does the diagram contain enough information to allow you to conclude that JK ML? Explain. 23. Write About It Draw a diagram and explain how a surveyor can set up triangles to find the distance across a lake. Label each part of your diagram. List which sides or angles must be congruent. 24. Which of these will NOT be used as a reason in a proof of AC AD? SAS ASA CPCTC Reflexive Property 25. Given the points K (1, 2), L (0, -4), M (-2, -3), and N (-1, 3), which of these is true? KNL MNL LNK NLM MLN KLN MNK NKL 26. What is the value of y? Which of these are NOT used to prove angles congruent? congruent triangles parallel lines noncorresponding parts perpendicular lines 264 Chapter 4 Triangle Congruence

54 28. Which set of coordinates represents the vertices of a triangle congruent to RST? (Hint: Find the lengths of the sides of RST.) (3, 4), (3, 0), (0, 0) (3, 1), (3, 3), (4, 6) (3, 3), (0, 4), (0, 0) (3, 0), (4, 4), (0, 6) CHALLENGE AND EXTEND 29. All of the edges of a cube are congruent. All of the angles on each face of a cube are right angles. Use CPCTC to explain why any two diagonals on the faces of a cube (for example, AC and AF ) must be congruent. 30. Given: JK ML, JM KL 31. Given: R is the midpoint of AB. Prove: J L S is the midpoint of DC. (Hint: Draw an auxiliary line.) RS AB, ASD BSC Prove: ASD BSC 32. ABC is in plane M. CDE is in plane P. Both planes have C in common and A E. What is the height AB to the nearest foot? SPIRAL REVIEW 33. Lina s test scores in her history class are 90, 84, 93, 88, and 91. What is the minimum score Lina must make on her next test to have an average test score of 90? (Previous course) 34. One long-distance phone plan costs $3.95 per month plus $0.08 per minute of use. A second long-distance plan costs $0.10 per minute for the first 50 minutes used each month and then $0.15 per minute after that. Which plan is cheaper if you use an average of 75 long-distance minutes per month? (Previous course) A figure has vertices at (1, 3), (2, 2), (3, 2), and (4, 3). Identify the transformation of the figure that produces an image with each set of vertices. (Lesson 1-7) 35. (1, -3), (2, -2), (3, -2), (4, -3) 36. (-2, -1), (-1, -2), (0, -2), (1, -1) 37. Determine if you can use ASA to prove ACB ECD. Explain. (Lesson 4-5) 4-6 Triangle Congruence: CPCTC 265

55 Quadratic Equations Algebra See Skills Bank page S66 Example A quadratic equation is an equation that can be written in the 2 form ax + bx + c = 0. California Standards 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Also covered: Review of 1A14.0, 1A20.0 Given: ABC is isosceles with AB AC. Solve for x. Step 1 Set x 2 5x equal to 6 to get x 2 5x = 6. Step 2 Rewrite the quadratic equation by subtracting 6 from each side to get x 2 5x 6 = 0. Step 3 Solve for x. Method 1: Factoring x 2-5x - 6 = 0 (x - 6) (x + 1) = 0 Factor. x - 6 = 0 or x + 1 = 0 x = 6 or x = -1 Solve. Set each factor equal to 0. Method 2: Quadratic Formula x = -b ± b 2-4ac 2a -(-5) ± (-5) 2-4 (1)(-6) x = Substitute 1 for 2 (1) a, -5 for b, and -6 for c. x = _ 5 ± 49 Simplify. 2 x = _ 5 ± 7 Find the square root. 2 x = _ 12 or x = _-2 Simplify. 2 2 x = 6 or x = -1 Step 4 Check each solution in the original equation. x 2-5x = 6 x 2-5x = 6 (6) 2-5 (6) 6 (-1) 2-5 (-1) Try This Solve for x in each isosceles triangle. 1. Given: FE FG 2. Given: JK JL 3. Given: YX YZ 4. Given: QP QR 266 Chapter 4 Triangle Congruence

56 4-7 Introduction to Coordinate Proof Objectives Position figures in the coordinate plane for use in coordinate proofs. Prove geometric concepts by using coordinate proof. Vocabulary coordinate proof California Standards 17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. Who uses this? The Bushmen in South Africa use the Global Positioning System to transmit data about endangered animals to conservationists. (See Exercise 24.) You have used coordinate geometry to find the midpoint of a line segment and to find the distance between two points. Coordinate geometry can also be used to prove conjectures. A coordinate proof is a style of proof that uses coordinate geometry and algebra. The first step of a coordinate proof is to position the given figure in the plane. You can use any position, but some strategies can make the steps of the proof simpler. Strategies for Positioning Figures in the Coordinate Plane Use the origin as a vertex, keeping the figure in Quadrant I. Center the figure at the origin. Center a side of the figure at the origin. Use one or both axes as sides of the figure. EXAMPLE 1 Positioning a Figure in the Coordinate Plane Position a rectangle with a length of 8 units and a width of 3 units in the coordinate plane. Method 1 You can center the longer side of the rectangle at the origin. Method 2 You can use the origin as a vertex of the rectangle. Depending on what you are using the figure to prove, one solution may be better than the other. For example, if you need to find the midpoint of the longer side, use the first solution. 1. Position a right triangle with leg lengths of 2 and 4 units in the coordinate plane. (Hint: Use the origin as the vertex of the right angle.) 4-7 Introduction to Coordinate Proof 267

57 Once the figure is placed in the coordinate plane, you can use slope, the coordinates of the vertices, the Distance Formula, or the Midpoint Formula to prove statements about the figure. EXAMPLE 2 Writing a Proof Using Coordinate Geometry Write a coordinate proof. Given: Right ABC has vertices A (0, 6), B (0, 0), and C (4, 0). D is the midpoint of AC. Prove: The area of DBC is one half the area of ABC. Proof: ABC is a right triangle with height AB and base BC. area of ABC = 1 2 bh = 1 (4)(6) = 12 square units 2 By the Midpoint Formula, the coordinates of D = ( , of DBC, and the base is 4 units. area of DBC = 1 Since 6 = ) = (2, 3). The y-coordinate of D is the height 2 bh = 1 (4)(3) = 6 square units 2 (12), the area of DBC is one half the area of ABC. 2. Use the information in Example 2 to write a coordinate proof showing that the area of ADB is one half the area of ABC. A coordinate proof can also be used to prove that a certain relationship is always true. You can prove that a statement is true for all right triangles without knowing the side lengths. To do this, assign variables as the coordinates of the vertices. EXAMPLE 3 Assigning Coordinates to Vertices Do not use both axes when positioning a figure unless you know the figure has a right angle. Position each figure in the coordinate plane and give the coordinates of each vertex. A a right triangle with leg B a rectangle with lengths a and b length c and width d 3. Position a square with side length 4p in the coordinate plane and give the coordinates of each vertex. If a coordinate proof requires calculations with fractions, choose coordinates that make the calculations simpler. For example, use multiples of 2 when you are to find coordinates of a midpoint. Once you have assigned the coordinates of the vertices, the procedure for the proof is the same, except that your calculations will involve variables. 268 Chapter 4 Triangle Congruence

58 EXAMPLE 4 Writing a Coordinate Proof Given: B is a right angle in ABC. D is the midpoint of AC. Prove: The area of DBC is one half the area of ABC. Because the x- and y-axes intersect at right angles, they can be used to form the sides of a right triangle. Step 1 Assign coordinates to each vertex. The coordinates of A are (0, 2 j), the coordinates of B are (0, 0), and the coordinates of C are (2n, 0). Step 2 Position the figure in the coordinate plane. Step 3 Write a coordinate proof. Proof: ABC is a right triangle with height 2j and base 2n. area of ABC = 1 2 bh = 1 (2n) 2 (2j) = 2nj square units Since you will use the Midpoint Formula to find the coordinates of D, use multiples of 2 for the leg lengths. By the Midpoint Formula, the coordinates of D = ( 0 + 2n The height of DBC is j units, and the base is 2n units. area of DBC = 1 2 bh = 1 (2n) 2 (j) = nj square units 2, 2j + 0 Since nj = 1 2(2nj), the area of DBC is one half the area of ABC. 2 ) = (n, j). 4. Use the information in Example 4 to write a coordinate proof showing that the area of ADB is one half the area of ABC. THINK AND DISCUSS 1. When writing a coordinate proof why are variables used instead of numbers as coordinates for the vertices of a figure? 2. How does the way you position a figure in the coordinate plane affect your calculations in a coordinate proof? 3. Explain why it might be useful to assign 2p as a coordinate instead of just p. 4. GET ORGANIZED Copy and complete the graphic organizer. In each row, draw an example of each strategy that might be used when positioning a figure for a coordinate proof. 4-7 Introduction to Coordinate Proof 269

59 4-7 Exercises GUIDED PRACTICE California Standards 5.0, 7.0, 17.0, 7AF1.0, 7AF2.0, 7MG2.1, 7MG3.2, 7MG3.4, 7MR2.3, 1A2.0, 1A Vocabulary What is the relationship between coordinate geometry, coordinate plane, and coordinate proof? KEYWORD: MG7 4-7 KEYWORD: MG7 Parent SEE EXAMPLE 1 p. 267 Position each figure in the coordinate plane. 2. a rectangle with a length of 4 units and width of 1 unit 3. a right triangle with leg lengths of 1 unit and 3 units SEE EXAMPLE 2 p. 268 SEE EXAMPLE 3 p. 268 Write a proof using coordinate geometry. 4. Given: Right PQR has coordinates P (0, 6), Q (8, 0), and R (0, 0). A is the midpoint of PR. B is the midpoint of QR. Prove: AB = 1 2 PQ Position each figure in the coordinate plane and give the coordinates of each vertex. 5. a right triangle with leg lengths m and n 6. a rectangle with length a and width b SEE EXAMPLE 4 p. 269 Multi-Step Assign coordinates to each vertex and write a coordinate proof. 7. Given: R is a right angle in PQR. A is the midpoint of PR. B is the midpoint of QR. Prove: AB = 1 2 PQ Independent Practice For See Exercises Example Extra Practice Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING Position each figure in the coordinate plane. 8. a square with side lengths of 2 units 9. a right triangle with leg lengths of 1 unit and 5 units Write a proof using coordinate geometry. 10. Given: Rectangle ABCD has coordinates A (0, 0), B (0, 10), C (6, 10), and D (6, 0). E is the midpoint of AB, and F is the midpoint of CD. Prove: EF = BC Position each figure in the coordinate plane and give the coordinates of each vertex. 11. a square with side length 2m 12. a rectangle with dimensions x and 3x Multi-Step Assign coordinates to each vertex and write a coordinate proof. 13. Given: E is the midpoint of AB in rectangle ABCD. F is the midpoint of CD. Prove: EF = AD 14. Critical Thinking Use variables to write the general form of the endpoints of a segment whose midpoint is (0, 0). 270 Chapter 4 Triangle Congruence

60 15. Recreation A hiking trail begins at E (0, 0). Bryan hikes from the start of the trail to a waterfall at W (3, 3) and then makes a 90 turn to a campsite at C (6, 0). a. Draw Bryan s route in the coordinate plane. b. If one grid unit represents 1 mile, what is the total distance Bryan hiked? Round to the nearest tenth. Find the perimeter and area of each figure. 16. a right triangle with leg lengths of a and 2a units 17. a rectangle with dimensions s and t units Find the missing coordinates for each figure Conservation 20. Conservation The Bushmen have sighted animals at the following coordinates: (-25, 31.5), (-23.2, 31.4), and (-24, 31.1). Prove that the distance between two of these locations is approximately twice the distance between two other. The origin of the springbok s name may come from its habit of pronking, or bouncing. When pronking, a springbok can leap up to 13 feet in the air. Springboks can run up to 53 miles per hour. 21. Navigation Two ships depart from a port at P (20, 10). The first ship travels to a location at A (-30, 50), and the second ship travels to a location at B (70, -30). Each unit represents one nautical mile. Find the distance to the nearest nautical mile between the two ships. Verify that the port is at the midpoint between the two. Write a coordinate proof. 22. Given: Rectangle PQRS has coordinates P (0, 2), Q (3, 2), R (3, 0), and S (0, 0). PR and QS intersect at T (1.5, 1). Prove: The area of RST is 1 of the area of the rectangle Given: A (x 1, y 1 ), B (x 2, y 2 ), with midpoint M ( x 1 + x 2, y 1 + y 2 2 Prove: AM = 1 2 AB 24. Plot the points on a coordinate plane and connect them to form KLM and MPK. Write a coordinate proof. Given: K (-2, 1), L (-2, 3), M (1, 3), P (1, 1) Prove: KLM MPK 25. Write About It When you place two sides of a figure on the coordinate axes, what are you assuming about the figure? 2 ) 26. This problem will prepare you for the Concept Connection on page 280. Paul designed a doghouse to fit against the side of his house. His plan consisted of a right triangle on top of a rectangle. a. Find BD and CE. b. Before building the doghouse, Paul sketched his plan on a coordinate plane. He placed A at the origin and AB on the x-axis. Find the coordinates of B, C, D, and E, assuming that each unit of the coordinate plane represents one inch. 4-7 Introduction to Coordinate Proof 271

61 27. The coordinates of the vertices of a right triangle are (0, 0), (4, 0), and (0, 2). Which is a true statement? The vertex of the right angle is at (4, 2). The midpoints of the two legs are at (2, 0) and (0, 1). The hypotenuse of the triangle is 6 units. The shortest side of the triangle is positioned on the x-axis. 28. A rectangle has dimensions of 2g and 2f units. If one vertex is at the origin, which coordinates could NOT represent another vertex? (2f, g) (2f, 0) (2g, 2f) (-2f, 2g) 29. The coordinates of the vertices of a rectangle are (0, 0), (a, 0), (a, b), and (0, b). What is the perimeter of the rectangle? a + b ab 1_ ab 2a + 2b A coordinate grid is placed over a map. City A is located at (-1, 2) and city C is located at (3, 5). If city C is at the midpoint between city A and city B, what are the coordinates of city B? (1, 3.5) (-5, -1) (7, 8) (2, 7) CHALLENGE AND EXTEND Find the missing coordinates for each figure The vertices of a right triangle are at (-2s, 2s), (0, 2s), and (0, 0). What coordinates could be used so that a coordinate proof would be easier to complete? 34. Rectangle ABCD has dimensions of 2f and 2g units. The equation of the line containing BD is y = g x, and f the equation of the line containing AC is y = - g f x + 2g. Use algebra to show that the coordinates of E are (f, g). SPIRAL REVIEW Use the quadratic formula to solve for x. Round to the nearest hundredth if necessary. (Previous course) = 8 x x = x 2 + 3x = 3 x 2 - x - 10 Find each value. (Lesson 3-2) 38. x 39. y 40. Use A (-4, 3), B (-1, 3), C (-3, 1), D (0, -2), E (3, -2), and F (2, -4) to prove ABC EDF. (Lesson 4-6). 272 Chapter 4 Triangle Congruence

62 4-8 Isosceles and Equilateral Triangles Objectives Prove theorems about isosceles and equilateral triangles. Apply properties of isosceles and equilateral triangles. Vocabulary legs of an isosceles triangle vertex angle base base angles Who uses this? Astronomers use geometric methods. (See Example 1.) Recall that an isosceles triangle has at least two congruent sides. The congruent sides are called the legs. The vertex angle is the angle formed by the legs. The side opposite the vertex angle is called the base, and the base angles are the two angles that have the base as a side. 3 is the vertex angle. 1 and 2 are the base angles. Theorems Isosceles Triangle California Standards 12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems. Also covered: 2.0, 4.0, 17.0 THEOREM HYPOTHESIS CONCLUSION Isosceles Triangle Theorem If two sides of a triangle are B C congruent, then the angles opposite the sides are congruent Converse of Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. DE DF Theorem is proven below. You will prove Theorem in Exercise 35. PROOF Isosceles Triangle Theorem Given: AB AC Prove: B C Proof: The Isosceles Triangle Theorem is sometimes stated as Base angles of an isosceles triangle are congruent. Statements 1. Draw X, the mdpt. of BC. 2. Draw the auxiliary line AX. 3. BX CX 4. AB AC 5. AX AX 6. ABX ACX 7. B C Reasons 1. Every seg. has a unique mdpt. 2. Through two pts. there is exactly one line. 3. Def. of mdpt. 4. Given 5. Reflex. Prop. of 6. SSS Steps 3, 4, 5 7. CPCTC 4-8 Isosceles and Equilateral Triangles 273

63 EXAMPLE 1 Astronomy Application The distance from Earth to nearby stars can be measured using the parallax method, which requires observing the positions of a star 6 months apart. If the distance LM to a star in July is km, explain why the distance LK to the star in January is the same. (Assume the distance from Earth to the Sun does not change.) m LKM = , so m LKM = Since LKM M, LMK is isosceles by the Converse of the Isosceles Triangle Theorem. Thus LK = LM = km. Not drawn to scale 1. If the distance from Earth to a star in September is km, what is the distance from Earth to the star in March? Explain. EXAMPLE 2 Finding the Measure of an Angle Find each angle measure. A m C m C = m B = x m C + m B + m A = 180 x + x + 38 = 180 2x = 142 x = 71 Thus m C = 71. B m S m S = m R Isosc. Thm. 2x = (x + 30) Substitute the given values. x = 30 Subtract x from both sides. Thus m S = 2x = 2 (30) = 60. Isosc. Thm. Sum Thm. Substitute the given values. Simplify and subtract 38 from both sides. Divide both sides by 2. Find each angle measure. 2a. m H 2b. m N The following corollary and its converse show the connection between equilateral triangles and equiangular triangles. Corollary Equilateral Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equilateral, then it is equiangular. (equilateral equiangular ) A B C You will prove Corollary in Exercise Chapter 4 Triangle Congruence

64 Corollary Equiangular Triangle COROLLARY HYPOTHESIS CONCLUSION If a triangle is equiangular, then it is equilateral. (equiangular equilateral ) DE DF EF You will prove Corollary in Exercise 37. EXAMPLE 3 Using Properties of Equilateral Triangles Find each value. A x ABC is equiangular. (3x + 15) = 60 3x = 45 x = 15 Equilateral equiangular The measure of each of an equiangular is 60. Subtract 15 from both sides. Divide both sides by 3. B t JKL is equilateral. 4t - 8 = 2t + 1 2t = 9 t = 4.5 Equiangular equilateral Def. of equilateral Subtract 2t and add 8 to both sides. Divide both sides by Use the diagram to find JL. EXAMPLE 4 Using Coordinate Proof A coordinate proof may be easier if you place one side of the triangle along the x-axis and locate a vertex at the origin or on the y-axis. Prove that the triangle whose vertices are the midpoints of the sides of an isosceles triangle is also isosceles. Given: ABC is isosceles. X is the mdpt. of AB. Y is the mdpt. of AC. Z is the mdpt. of BC. Prove: XYZ is isosceles. Proof: Draw a diagram and place the coordinates of ABC and XYZ as shown. By the Midpoint Formula, the coordinates of X are ( 2a + 0, 2b ) = (a, b), the coordinates of Y are 2a + 4a (, 2b ) = (3a, b), and the coordinates of Z are ( 4a + 0, ) = (2a, 0). By the Distance Formula, XZ = (2a - a) 2 + (0 - b) 2 = a 2 + b 2, and YZ = (2a - 3a) 2 + (0 - b) 2 = a 2 + b 2. Since XZ = YZ, XZ YZ by definition. So XYZ is isosceles. 4. What if...? The coordinates of ABC are A (0, 2b), B (-2a, 0), and C (2a, 0). Prove XYZ is isosceles. 4-8 Isosceles and Equilateral Triangles 275

65 THINK AND DISCUSS 1. Explain why each of the angles in an equilateral triangle measures GET ORGANIZED Copy and complete the graphic organizer. In each box, draw and mark a diagram for each type of triangle. 4-8 California Standards Exercises 2.0, 4.0, 17.0, KEYWORD: MG7 4-8 GUIDED PRACTICE 7AF4.1, 7MG3.4, 7MR1.2, 7MR2.3, 1A2.0 KEYWORD: MG7 Parent 1. Vocabulary Draw isosceles JKL with K as the vertex angle. Name the legs, base, and base angles of the triangle. SEE EXAMPLE 1 p Surveying To find the distance QR across a river, a surveyor locates three points Q, R, and S. QS = 41 m, and m S = 35. The measure of exterior PQS = 70. Draw a diagram and explain how you can find QR. SEE EXAMPLE 2 p. 274 Find each angle measure. 3. m ECD 4. m K SEE EXAMPLE 3 p m X 6. m A Find each value. 7. y 8. x 9. BC 10. JK SEE EXAMPLE 4 p Given: ABC is right isosceles. X is the midpoint of AC. AB BC Prove: AXB is isosceles. 276 Chapter 4 Triangle Congruence

66 Independent Practice For See Exercises Example Extra Practice Skills Practice p. S11 Application Practice p. S31 PRACTICE AND PROBLEM SOLVING 12. Aviation A plane is flying parallel to the ground along AC. When the plane is at A, an air-traffic controller in tower T measures the angle to the plane as 40. After the plane has traveled 2.4 mi to B, the angle to the plane is 80. How can you find BT? Find each angle measure. 13. m E 14. m TRU 15. m F 16. m A Find each value. 17. z 18. y 19. BC 20. XZ 21. Given: ABC is isosceles. P is the midpoint of AB. Q is the midpoint of AC. AB AC Prove: PC QB Tell whether each statement is sometimes, always, or never true. Support your answer with a sketch. 22. An equilateral triangle is an isosceles triangle. 23. The vertex angle of an isosceles triangle is congruent to the base angles. 24. An isosceles triangle is a right triangle. 25. An equilateral triangle and an obtuse triangle are congruent. 26. Critical Thinking Can a base angle of an isosceles triangle be an obtuse angle? Why or why not? 4-8 Isosceles and Equilateral Triangles 277

67 27. This problem will prepare you for the Concept Connection page 280. The diagram shows the inside view of the support structure of the back of a doghouse. PQ PR, PS PT, m PST = 71, and m QPS = m RPT = 18. a. Find m SPT. b. Find m PQR and m PRQ. Multi-Step Find the measure of each numbered angle Write a coordinate proof. Given: B is a right angle in isosceles right ABC. X is the midpoint of AC. BA BC Prove: AXB CXB 31. Estimation Draw the figure formed by (-2, 1), (5, 5), and (-1, -7). Estimate the measure of each angle and make a conjecture about the classification of the figure. Then use a protractor to measure each angle. Was your conjecture correct? Why or why not? 32. How many different isosceles triangles have a perimeter of 18 and sides whose lengths are natural numbers? Explain. Multi-Step Find the value of the variable in each diagram Navigation The taffrail log is dragged from the stern of a vessel to measure the speed or distance traveled during a voyage. The log consists of a rotator, recording device, and governor. 35. Prove the Converse of the Isosceles Triangle Theorem. 36. Complete the proof of Corollary Given: AB AC BC Prove: A B C Proof: Since AB AC, a.? by the Isosceles Triangle Theorem. Since AC BC, A B by b.?. Therefore A C by c.?. By the Transitive Property of, A B C. 37. Prove Corollary Navigation The captain of a ship traveling along AB sights an island C at an angle of 45. The captain measures the distance the ship covers until it reaches B, where the angle to the island is 90. Explain how to find the distance BC to the island. 39. Given: ABC CBA Prove: ABC is isosceles. 40. Write About It Write the Isosceles Triangle Theorem and its converse as a biconditional. 278 Chapter 4 Triangle Congruence

68 41. Rewrite the paragraph proof of the Hypotenuse-Leg (HL) Congruence Theorem as a two-column proof. Given: ABC and DEF are right triangles. C and F are right angles. AC DF, and AB DE. Prove: ABC DEF Proof: On DEF draw EF. Mark G so that FG = CB. Thus FG CB. From the diagram, AC DF and C and F are right angles. DF EG by definition of perpendicular lines. Thus DFG is a right angle, and DFG C. ABC DGF by SAS. DG AB by CPCTC. AB DE as given. DG DE by the Transitive Property. By the Isosceles Triangle Theorem G E. DFG DFE since right angles are congruent. So DGF DEF by AAS. Therefore ABC DEF by the Transitive Property. 42. Lorena is designing a window so that R, S, T, and U are right angles, VU VT, and m UVT = 20. What is m RUV? Which of these values of y makes ABC isosceles? 1 _ 1 7 _ _ _ Gridded Response The vertex angle of an isosceles triangle measures (6t - 9), and one of the base angles measures (4t). Find t. CHALLENGE AND EXTEND 45. In the figure, JK JL, and KM KL. Let m J = x. Prove m MKL must also be x. 46. An equilateral ABC is placed on a coordinate plane. Each side length measures 2a. B is at the origin, and C is at (2a, 0). Find the coordinates of A. 47. An isosceles triangle has coordinates A (0, 0) and B (a, b). What are all possible coordinates of the third vertex? SPIRAL REVIEW Find the solutions for each equation. (Previous course) 48. x 2 + 5x + 4 = x 2-4x + 3 = x 2-2x + 1 = 0 Find the slope of the line that passes through each pair of points. (Lesson 3-5) 51. (2, -1) and (0, 5) 52. (-5, -10) and (20, -10) 53. (4, 7) and (10, 11) 54. Position a square with a perimeter of 4s in the coordinate plane and give the coordinates of each vertex. (Lesson 4-7) 4-8 Isosceles and Equilateral Triangles 279

69 SECTION 4B Proving Triangles Congruent Gone to the Dogs You are planning to build a doghouse for your dog. The pitched roof of the doghouse will be supported by four trusses. Each truss will be an isosceles triangle with the dimensions shown. To determine the materials you need to purchase and how you will construct the trusses, you must first plan carefully. 1. You want to be sure that all four trusses are exactly the same size and shape. Explain how you could measure three lengths on each truss to ensure this. Which postulate or theorem are you using? 2. Prove that the two triangular halves of the truss are congruent. 3. What can you say about AD and DB? Why is this true? Use this to help you find the lengths of AD, DB, AC, and BC. 4. You want to make careful plans on a coordinate plane before you begin your construction of the trusses. Each unit of the coordinate plane represents 1 inch. How could you assign coordinates to vertices A, B, and C? 5. m ACB = 106. What is the measure of each of the acute angles in the truss? Explain how you found your answer. 6. You can buy the wood for the trusses at the building supply store for $0.80 a foot. The store sells the wood in 6-foot lengths only. How much will you have to spend to get enough wood for the 4 trusses of the doghouse? 280 Chapter 4 Triangle Congruence

70 Quiz for Lessons 4-4 Through Triangle Congruence: SSS and SAS 1. The figure shows one tower and the cables of a suspension bridge. Given that AC BC, use SAS to explain why ACD BCD. 2. Given: JK bisects MJN. MJ NJ Prove: MJK NJK SECTION 4B 4-5 Triangle Congruence: ASA, AAS, and HL Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know. 3. RSU and TUS 4. ABC and DCB Observers in two lighthouses K and L spot a ship S. 5. Draw a diagram of the triangle formed by the lighthouses and the ship. Label each measure. 6. Is there enough data in the table to pinpoint the location of the ship? Why? K to L K to S L to S Bearing E N 58 E N 77 W Distance 12 km?? 4-6 Triangle Congruence: CPCTC 7. Given: CD BE, DE CB Prove: D B 4-7 Introduction to Coordinate Proof 8. Position a square with side lengths of 9 units in the coordinate plane 9. Assign coordinates to each vertex and write a coordinate proof. Given: ABCD is a rectangle with M as the midpoint of AB. N is the midpoint of AD. Prove: The area of AMN is 1 the area of rectangle ABCD Isosceles and Equilateral Triangles Find each value. 10. m C 11. ST 12. Given: Isosceles JKL has coordinates J (0, 0), K (2a, 2b), and L (4a, 0). M is the midpoint of JK, and N is the midpoint of KL. Prove: KMN is isosceles. Ready to Go On? 281

71 EXTENSION Proving Constructions Valid Objective Use congruent triangles to prove constructions valid. When performing a compass and straight edge construction, the compass setting remains the same width until you change it. This fact allows you to construct a segment congruent to a given segment. You can assume that two distances constructed with the same compass setting are congruent. California Standards 2.0 Students write geometric proofs, including proofs by contradiction. Also covered: 5.0 The steps in the construction of a figure can be justified by combining the assumptions of compass and straightedge constructions and the postulates and theorems that are used for proving triangles congruent. You have learned that there exists exactly one midpoint on any line segment. The proof below justifies the construction of a midpoint. EXAMPLE 1 Proving the Construction of a Midpoint Given: diagram showing the steps in the construction Prove: M is the midpoint of AB. To construct a midpoint, see the construction of a perpendicular bisector on p Proof: Statements 1. Draw AC, BC, AD, and BD. 2. AC BC AD BD 3. CD CD 4. ACD BCD 5. ACD BCD 6. CM CM 7. ACM BCM 8. AM BM 9. M is the midpt. of AB. Reasons 1. Through any two pts. there is exactly one line. 2. Same compass setting used 3. Reflex. Prop. of 4. SSS Steps 2, 3 5. CPCTC 6. Reflex. Prop. of 7. SAS Steps 2, 5, 6 8. CPCTC 9. Def. of mdpt. 1. Given: above diagram Prove: CD is the perpendicular bisector of AB. 282 Chapter 4 Triangle Congruence

72 EXAMPLE 2 Proving the Construction of an Angle Given: diagram showing the steps in the construction Prove: A D To review the construction of an angle congruent to another angle, see page 22. Proof: Since there is a straight line through any two points, you can draw BC and EF. The same compass setting was used to construct AC, AB, DF, and DE, so AC AB DF DE. The same compass setting was used to construct BC and EF, so BC EF. Therefore BAC EDF by SSS, and A D by CPCTC. 2. Prove the construction for bisecting an angle. (See page 23.) EXTENSION Exercises Use each diagram to prove the construction valid. 1. parallel lines 2. a perpendicular through a point not (See page 163 and page 170.) on the line (See page 179.) 3. constructing a triangle using SAS 4. constructing a triangle using ASA (See page 243.) (See page 253.) Extension 283

73 Vocabulary acute triangle auxiliary line base base angle congruent polygons coordinate proof corollary corresponding angles corresponding sides CPCTC equiangular triangle equilateral triangle exterior exterior angle included angle included side interior interior angle For a complete list of the postulates and theorems in this chapter, see p. S82. isosceles triangle legs of an isosceles triangle obtuse triangle remote interior angle right triangle scalene triangle triangle rigidity vertex angle Complete the sentences below with vocabulary words from the list above. 1. A(n)? is a triangle with at least two congruent sides. 2. A name given to matching angles of congruent triangles is?. 3. A(n)? is the common side of two consecutive angles in a polygon. 4-1 Classifying Triangles (pp ) 12.0 EXAMPLE Classify the triangle by its angle measures and side lengths. isosceles right triangle EXERCISES Classify each triangle by its angle measures and side lengths Angle Relationships in Triangles (pp ) 2.0, 12.0, 13.0 EXAMPLE Find m S. 12x = 3x x 12x = 9x x = 42 x = 14 m S = 6 (14) = 84 EXERCISES Find m N In LMN, m L = 8x, m M = (2x + 1), and m N = (6x - 1). 284 Chapter 4 Triangle Congruence

74 4-3 Congruent Triangles (pp ) 2.0, 5.0 EXAMPLE Given: DEF JKL. Identify all pairs of congruent corresponding parts. Then find the value of x. The congruent pairs follow: D J, E K, F L, DE JK, EF KL, and DF JL. Since m E = m K, 90 = 8x After 22 is added to both sides, 112 = 8x. So x = 14. EXERCISES Given: PQR XYZ. Identify the congruent corresponding parts. 8. PR? 9. Y? Given: ABC CDA Find each value. 10. x 11. CD 4-4 Triangle Congruence: SSS and SAS (pp ) 2.0, 5.0, 16.0 EXAMPLES Given: RS UT, and VS VT. V is the midpoint of RU. Prove: RSV UTV Proof: Statements 1. RS UT 2. VS VT 3. V is the mdpt. of RU. 4. RV UV 5. RSV UTV Reasons 1. Given 2. Given 3. Given 4. Def. of mdpt. 5. SSS Steps 1, 2, 4 Show that ADB CDB when s = 5. AB = s 2-4s AD = 14-2s = (5) = 14-2 (5) = 5 = 4 BD BD by the Reflexive Property. AD CD and AB CB. So ADB CDB by SSS. EXERCISES 12. Given: AB DE, DB AE Prove: ADB DAE 13. Given: GJ bisects FH, and FH bisects GJ. Prove: FGK HJK 14. Show that ABC XYZ when x = Show that LMN PQR when y = 25. Study Guide: Review 285

75 4-5 Triangle Congruence: ASA, AAS, and HL (pp ) 2.0, 4.0, 5.0, 16.0 EXAMPLES Given: B is the midpoint of AE. A E, ABC EBD Prove: ABC EBD EXERCISES 16. Given: C is the midpoint of AG. HA GB Prove: HAC BGC Proof: Statements 1. A E 2. ABC EBD 3. B is the mdpt. of AE. 4. AB EB 5. ABC EBD Reasons 1. Given 2. Given 3. Given 4. Def. of mdpt. 5. ASA Steps 1, 4, Given: WX XZ, YZ ZX, WZ YX Prove: WZX YXZ 18. Given: S and V are right angles. RT = UW. m T = m W Prove: RST UVW 4-6 Triangle Congruence: CPCTC (pp ) 2.0, 5.0 EXAMPLES Given: JL and HK bisect each other. Prove: JHG LKG Proof: Statements 1. JL and HK bisect each other. 2. JG LG, and HG KG. 3. JGH LGK 4. JHG LKG 5. JHG LKG 1. Given Reasons 2. Def. of bisect 3. Vert. Thm. 4. SAS Steps 2, 3 5. CPCTC EXERCISES 19. Given: M is the midpoint of BD. BC DC Prove: Given: PQ RQ, PS RS Prove: QS bisects PQR. 21. Given: H is the midpoint of GJ. L is the midpoint of MK. GM KJ, GJ KM, G K Prove: GMH KJL 286 Chapter 4 Triangle Congruence

76 4-7 Introduction to Coordinate Proof (pp ) 17.0 EXAMPLES Given: B is a right angle in isosceles right ABC. E is the midpoint of AB. D is the midpoint of CB. AB CB Prove: CE AD Proof: Use the coordinates A(0, 2a), B(0, 0), and C (2a, 0). Draw AD and CE. By the Midpoint Formula, E = ( _ , _ 2a ) = (0, a) and D = ( _ 0 + 2a,_ ) = (a, 0) By the Distance Formula, CE = (2a - 0) 2 + (0 - a) 2 = 4a 2 + a 2 = a 5 AD = (a - 0) 2 + (0-2a) 2 = a 2 + 4a 2 = a 5 Thus CE AD by the definition of congruence. EXERCISES Position each figure in the coordinate plane and give the coordinates of each vertex. 22. a right triangle with leg lengths r and s 23. a rectangle with length 2p and width p 24. a square with side length 8m For exercises 25 and 26 assign coordinates to each vertex and write a coordinate proof. 25. Given: In rectangle ABCD, E is the midpoint of AB, F is the midpoint of BC, G is the midpoint of CD, and H is the midpoint of AD. Prove: EF GH 26. Given: PQR has a right Q. M is the midpoint of PR. Prove: MP = MQ = MR 27. Show that a triangle with vertices at (3, 5), (3, 2), and (2, 5) is a right triangle. 4-8 Isosceles and Equilateral Triangles (pp ) 2.0, 4.0, 12.0, 17.0 EXAMPLE Find the value of x. m D + m E + m F = 180 by the Triangle Sum Theorem. m E = m F by the Isosceles Triangle Theorem. m D + 2 m E = 180 Substitution (3x) = 180 Substitute the given values. 6x = 138 x = 23 Simplify. Divide both sides by 6. EXERCISES Find each value. 28. x 29. RS 30. Given: ACD is isosceles with D as the vertex angle. B is the midpoint of AC. AB = x + 5, BC = 2x - 3, and CD = 2x + 6. Find the perimeter of ACD. Study Guide: Review 287

77 1. Classify ACD by its angle measures. x Ç Classify each triangle by its side lengths ACD 4. ABC x ABD Î, 5. While surveying the triangular plot of land shown, a surveyor finds that m S = 43. The measure of RTP is twice that of RTS. What is m R? Given: XYZ JKL Identify the congruent corresponding parts. 6. JL? 7. Y? 10. Given: T is the midpoint of PR and SQ. Prove: PTS RTQ {ÎÂ - / 8. L * 9. YZ?? * / - +, 11. The figure represents a walkway with triangular supports. Given that GJ bisects HGK and H K, use AAS to prove HGJ KGJ 12. Given: AB DC, AB AC, DC DB Prove: ABC DCB 13. Given: PQ SR, S Q Prove: PS QR * + -, 14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex. 15. Assign coordinates to each vertex and write a coordinate proof. Given: Square ABCD Prove: AC BD Find each value. 16. y 17. m S - * xèâ xê Ê Þ Â, Given: Isosceles ABC has coordinates A(2a, 0), B(0, 2b), and C(-2a, 0). D is the midpoint of AC, and E is the midpoint of AB. Prove: AED is isosceles. 288 Chapter 4 Triangle Congruence /